compare:
i can use some of the same methods to multiply both:
- a table
- the __________ property
- vertical arrangement
contrast:
- i have to multiply more terms when i multiply __________.
- for __________, i have to add three partial products instead of two.
- i can use foil for __________, but not for __________
find the product. show your work.
1. $(4x^2 - 3x + 2)(3x^2 + 4x + 1)$
\ \ \ \ \ \ $\begin{array}{r}
4x^2 - 3x + 2\\
\times\ 3x^2 + 4x + 1\\
\hline
\end{array}$

Question

Accepted Answer

### Compare:
#### Step 1: Identify the property for multiplication
The distributive property (also known as the distributive property of multiplication over addition) is commonly used when multiplying polynomials. So the blank here should be "distributive".

### Contrast:
1. When multiplying a trinomial by a trinomial (or a polynomial with more terms), we have to multiply more terms compared to multiplying a binomial by a binomial. So the first blank: "a trinomial by a trinomial" (or similar, e.g., "two trinomials").
2. For multiplying a trinomial by a trinomial, we have three partial products (since each term in the first trinomial multiplies each term in the second trinomial, and a trinomial has three terms), whereas a binomial by a binomial has two partial products. So the second blank: "multiplying a trinomial by a trinomial".
3. FOIL (First, Outer, Inner, Last) is used for multiplying two binomials (since a binomial has two terms). It does not work for multiplying trinomials (or polynomials with more than two terms). So the third blank: "binomials", and the fourth blank: "trinomials" (or "polynomials with more than two terms").

### Find the product of $(4x^{2}-3x + 2)(3x^{2}+4x + 1)$
#### Step 1: Use the distributive property (also called the "FOIL - like" extension for trinomials, or simply distributing each term in the first polynomial to each term in the second polynomial)
We multiply $4x^{2}$ by each term in $(3x^{2}+4x + 1)$, then $-3x$ by each term in $(3x^{2}+4x + 1)$, and then $2$ by each term in $(3x^{2}+4x + 1)$, and then combine like terms.

- Multiply $4x^{2}$ with $(3x^{2}+4x + 1)$:
$4x^{2}\times3x^{2}=12x^{4}$, $4x^{2}\times4x = 16x^{3}$, $4x^{2}\times1=4x^{2}$
- Multiply $-3x$ with $(3x^{2}+4x + 1)$:
$-3x\times3x^{2}=-9x^{3}$, $-3x\times4x=-12x^{2}$, $-3x\times1 = - 3x$
- Multiply $2$ with $(3x^{2}+4x + 1)$:
$2\times3x^{2}=6x^{2}$, $2\times4x = 8x$, $2\times1=2$

#### Step 2: Combine like terms
- For $x^{4}$ terms: $12x^{4}$ (only one term)
- For $x^{3}$ terms: $16x^{3}-9x^{3}=7x^{3}$
- For $x^{2}$ terms: $4x^{2}-12x^{2}+6x^{2}=-2x^{2}$
- For $x$ terms: $-3x + 8x=5x$
- Constant term: $2$ (only one term)

Now, combine all these results:
$12x^{4}+7x^{3}-2x^{2}+5x + 2$

### Final Answers for the Blanks (Compare and Contrast):
- Compare: the $\boldsymbol{\text{distributive}}$ property
- Contrast 1: I have to multiply more terms when I multiply $\boldsymbol{\text{a trinomial by a trinomial}}$ (or "two trinomials")
- Contrast 2: For $\boldsymbol{\text{multiplying a trinomial by a trinomial}}$, I have to add three partial products instead of two.
- Contrast 3: I can use FOIL for $\boldsymbol{\text{binomials}}$ but not for $\boldsymbol{\text{trinomials}}$ (or "polynomials with more than two terms")

### Product of $(4x^{2}-3x + 2)(3x^{2}+4x + 1)$
$\boldsymbol{12x^{4}+7x^{3}-2x^{2}+5x + 2}$

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QUESTION IMAGE

Question

Explanation:

Compare:

Step 1: Identify the property for multiplication

Contrast:

Find the product of \((4x^{2}-3x + 2)(3x^{2}+4x + 1)\)

Step 1: Use the distributive property (also called the "FOIL - like" extension for trinomials, or simply distributing each term in the first polynomial to each term in the second polynomial)

Step 2: Combine like terms

Final Answers for the Blanks (Compare and Contrast):

Product of \((4x^{2}-3x + 2)(3x^{2}+4x + 1)\)

Answer: