simplify the expression $3x(x - 12x) + 3x^2 - 2(x - 2)^2$. which statements are true about the process and simplified product? choose three correct answers.
- the final simplified product is $-28x^2 + 8x - 8$.
- after multiplying, the like terms are combined by adding and subtracting.
- the parentheses are eliminated through multiplication.
- the simplified product is a binomial.
- the term $-2(x - 2)^2$ is simplified by first squaring the expression $x - 2$.

Question

Accepted Answer

To solve this, we first simplify the given expression $3x(x - 12x)+3x^{2}-2(x - 2)^{2}$ step by step:

### Step 1: Simplify inside the first parentheses and expand the square
- Simplify $x - 12x$: $x-12x=- 11x$
- Expand $(x - 2)^{2}$ using the formula $(a - b)^{2}=a^{2}-2ab + b^{2}$, where $a = x$ and $b = 2$. So $(x - 2)^{2}=x^{2}-4x + 4$

### Step 2: Multiply to eliminate parentheses
- Multiply $3x$ with $-11x$: $3x\times(-11x)=-33x^{2}$
- Multiply $-2$ with $(x^{2}-4x + 4)$: $-2(x^{2}-4x + 4)=-2x^{2}+8x - 8$

### Step 3: Combine like terms
- The original expression becomes: $-33x^{2}+3x^{2}-2x^{2}+8x - 8$
- Combine the $x^{2}$ terms: $-33x^{2}+3x^{2}-2x^{2}=(-33 + 3-2)x^{2}=-32x^{2}$? Wait, no, wait, let's re - check:

Wait, the original expression is $3x(x - 12x)+3x^{2}-2(x - 2)^{2}$

First, $x-12x=-11x$, so $3x(x - 12x)=3x\times(-11x)=-33x^{2}$

Then $3x(x - 12x)+3x^{2}=-33x^{2}+3x^{2}=-30x^{2}$

Then we have $-30x^{2}-2(x - 2)^{2}$

Expand $(x - 2)^{2}=x^{2}-4x + 4$, so $-2(x - 2)^{2}=-2x^{2}+8x - 8$

Now combine with $-30x^{2}$: $-30x^{2}-2x^{2}+8x - 8=-32x^{2}+8x - 8$. Wait, there must be a miscalculation. Wait, the first term: $3x(x - 12x)=3x\times(-11x)=-33x^{2}$, then $+3x^{2}$ gives $-33x^{2}+3x^{2}=-30x^{2}$, then $-2(x^{2}-4x + 4)=-2x^{2}+8x - 8$. So total $x^{2}$ terms: $-30x^{2}-2x^{2}=-32x^{2}$, and the linear term $8x$ and constant term $-8$. But the option says $-28x^{2}+8x - 8$. So maybe my initial step is wrong.

Wait, maybe the first parenthesis is $x-12$ instead of $x - 12x$? If it is $3x(x - 12)+3x^{2}-2(x - 2)^{2}$

Then $3x(x - 12)=3x^{2}-36x$

$3x^{2}-36x+3x^{2}=6x^{2}-36x$

$-2(x - 2)^{2}=-2(x^{2}-4x + 4)=-2x^{2}+8x - 8$

Now combine: $6x^{2}-36x-2x^{2}+8x - 8 = 4x^{2}-28x - 8$. No, that's not matching.

Wait, let's go back to the original problem. The user's problem is $3x(x - 12x)+3x^{2}-2(x - 2)^{2}$

Let's recalculate:

$x-12x=-11x$, so $3x(x - 12x)=3x\times(-11x)=-33x^{2}$

$3x(x - 12x)+3x^{2}=-33x^{2}+3x^{2}=-30x^{2}$

$(x - 2)^{2}=x^{2}-4x + 4$, so $-2(x - 2)^{2}=-2x^{2}+8x - 8$

Now, $-30x^{2}-2x^{2}+8x - 8=-32x^{2}+8x - 8$. But the option has $-28x^{2}+8x - 8$. So there is a mistake in my calculation or in the problem statement. But let's analyze the options:

1. **"The final simplified product is $-28x^{2}+8x - 8$"**: Let's assume that maybe the first term is $3x(x - 8x)$ instead of $3x(x - 12x)$. If $3x(x - 8x)=3x\times(-7x)=-21x^{2}$, then $-21x^{2}+3x^{2}=-18x^{2}$, then $-2(x^{2}-4x + 4)=-2x^{2}+8x - 8$, then $-18x^{2}-2x^{2}+8x - 8=-20x^{2}+8x - 8$. No. Alternatively, maybe the first term is $3x(x - 10x)$: $3x\times(-9x)=-27x^{2}$, $+3x^{2}=-24x^{2}$, $-2x^{2}=-26x^{2}$. Not matching.

But let's analyze the options:

- **"After multiplying, the like terms are combined by adding and subtracting."**: This is true. When we simplify algebraic expressions, after expanding (multiplying to remove parentheses), we combine like terms by adding or subtracting their coefficients.
- **"The parentheses are eliminated through multiplication."**: To remove parentheses in expressions like $a(b + c)$ or $(a + b)^{2}$ (which is $(a + b)(a + b)$), we use the distributive property (multiplication). So this is true.
- **"The term $-2(x - 2)^{2}$ is simplified by first squaring the expression $x - 2$."**: To simplify $-2(x - 2)^{2}$, we first expand $(x - 2)^{2}$ (by squaring $x - 2$) and then multiply by $-2$. So this is true.
- **"The final simplified product is $-28x^{2}+8x - 8$."**: Let's recalculate the expression correctly. Let's assume that the first term is $3x(x - 8)+3x^{2}-2(x - 2)^{2}$

$3x(x - 8)=3x^{2}-24x$

$3x^{2}-24x + 3x^{2}=6x^{2}-24x$

$-2(x - 2)^{2}=-2(x^{2}-4x + 4)=-2x^{2}+8x - 8$

$6x^{2}-24x-2x^{2}+8x - 8 = 4x^{2}-16x - 8$. No.

Wait, maybe the original expression is $3x(x - 12x)+3x^{2}-2(x + 2)^{2}$

$(x + 2)^{2}=x^{2}+4x + 4$, $-2(x + 2)^{2}=-2x^{2}-8x - 8$

$3x(x - 12x)=-33x^{2}$, $-33x^{2}+3x^{2}=-30x^{2}$, $-30x^{2}-2x^{2}-8x - 8=-32x^{2}-8x - 8$. No.

Alternatively, maybe the first term is $3x(12x - x)$ (i.e., $3x(11x)$): $33x^{2}$, $+3x^{2}=36x^{2}$, $-2(x - 2)^{2}=-2x^{2}+8x - 8$, $36x^{2}-2x^{2}+8x - 8 = 34x^{2}+8x - 8$. No.

But regardless of the miscalculation in the final product, let's analyze the options:

1. "After multiplying, the like terms are combined by adding and subtracting." - True. When simplifying algebraic expressions, after using the distributive property (multiplication) to remove parentheses, we combine like terms by adding or subtracting their coefficients.
2. "The parentheses are eliminated through multiplication." - True. To remove parentheses in expressions like $a(b + c)$ or $(a + b)^{2}$ (which is a product of $(a + b)$ and $(a + b)$), we use multiplication (distributive property for the first case and squaring - which is a form of multiplication for the second case).
3. "The term $-2(x - 2)^{2}$ is simplified by first squaring the expression $x - 2$." - True. To simplify $-2(x - 2)^{2}$, we first calculate $(x - 2)^{2}=x^{2}-4x + 4$ (by squaring $x - 2$) and then multiply by $-2$.
4. "The final simplified product is $-28x^{2}+8x - 8$." - Let's assume that there was a typo and the first term is $3x(x - 10x)$: $3x\times(-9x)=-27x^{2}$, $+3x^{2}=-24x^{2}$, $-2(x - 2)^{2}=-2x^{2}+8x - 8$, $-24x^{2}-2x^{2}+8x - 8=-26x^{2}+8x - 8$. Still not $-28x^{2}$. If the first term is $3x(x - 8x)$: $3x\times(-7x)=-21x^{2}$, $+3x^{2}=-18x^{2}$, $-2x^{2}=-20x^{2}$. No. If the first term is $3x(12x - x)$ (i.e., $3x(11x)$): $33x^{2}$, $+3x^{2}=36x^{2}$, $-2(x - 2)^{2}=-2x^{2}+8x - 8$, $36x^{2}-2x^{2}+8x - 8 = 34x^{2}+8x - 8$. No.

5. "The simplified product is a binomial." - A binomial has two terms. The simplified product (if we assume the correct final form has three terms: quadratic, linear, and constant) is a trinomial, so this is false.

So the three correct statements are:

- After multiplying, the like terms are combined by adding and subtracting.
- The parentheses are eliminated through multiplication.
- The term $-2(x - 2)^{2}$ is simplified by first squaring the expression $x - 2$.

(Note: There might be a typo in the problem statement regarding the final product, but based on the process of simplifying algebraic expressions, these three statements are correct.)

# Brief Explanations:
- "After multiplying, the like terms are combined by adding and subtracting": When simplifying algebraic expressions, after expanding (multiplying to remove parentheses), we combine like terms by adding or subtracting their coefficients.
- "The parentheses are eliminated through multiplication": To remove parentheses in expressions like $a(b + c)$ or $(a + b)^2$ (a product of $(a + b)$ and $(a + b)$), we use the distributive property (multiplication).
- "The term $-2(x - 2)^2$ is simplified by first squaring the expression $x - 2$": To simplify $-2(x - 2)^2$, we first expand $(x - 2)^2$ (by squaring $x - 2$) and then multiply by $-2$.
# Answer:
- After multiplying, the like terms are combined by adding and subtracting.
- The parentheses are eliminated through multiplication.
- The term $-2(x - 2)^{2}$ is simplified by first squaring the expression $x - 2$.

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QUESTION IMAGE

Question

Explanation:

Step 1: Simplify inside the first parentheses and expand the square

Step 2: Multiply to eliminate parentheses

Step 3: Combine like terms

Answer: