Question

\\((x^2 + 4x + 3)^2\\)

Explanation:

Identify the expression and method

We want to expand the squared trinomial:

$$ (x^2 + 4x + 3)^2 $$

We can use the Trinomial Squaring formula:

$$ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca $$

Let's define our terms:

• \(a = x^2\)
• \(b = 4x\)
• \(c = 3\)

Square the individual terms

Using Polynomial Multiplication, we square each term:

• \(a^2 = (x^2)^2 = x^4\)
• \(b^2 = (4x)^2 = 16x^2\)
• \(c^2 = (3)^2 = 9\)

Calculate the cross-product terms

Next, we find the twice-product terms:

• \(2ab = 2(x^2)(4x) = 8x^3\)
• \(2bc = 2(4x)(3) = 24x\)
• \(2ca = 2(3)(x^2) = 6x^2\)

Combine all terms

Now, we write out the sum of all these parts:

$$ x^4 + 16x^2 + 9 + 8x^3 + 24x + 6x^2 $$

Simplify and arrange in descending order

Using Polynomial Simplification, we group and combine the like terms (\(16x^2\) and \(6x^2\)):

$$ 16x^2 + 6x^2 = 22x^2 $$

Arranging the terms from highest degree to lowest degree:

$$ x^4 + 8x^3 + 22x^2 + 24x + 9 $$

Answer:

$$x^4 + 8x^3 + 22x^2 + 24x + 9$$