High School

Multiply:

[tex]\[

\begin{array}{r}

x^2 + 4x + 2 \\

\times \quad 2x^2 + 3x - 4 \\

\hline

\end{array}

\][/tex]

A. [tex]\(2x^4 + 12x^2 - 8\)[/tex]

B. [tex]\(2x^4 + 23x^2 - 10x - 8\)[/tex]

C. [tex]\(3x^2 + 7x - 2\)[/tex]

D. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex]

Answer :

To multiply the polynomials [tex]\((x^2 + 4x + 2)\)[/tex] and [tex]\((2x^2 + 3x - 4)\)[/tex], we'll use the distributive property, also known as the FOIL method for distributing across multiple terms.

1. Multiply each term in the first polynomial by each term in the second polynomial:

- [tex]\(x^2 \times 2x^2 = 2x^4\)[/tex]
- [tex]\(x^2 \times 3x = 3x^3\)[/tex]
- [tex]\(x^2 \times (-4) = -4x^2\)[/tex]

- [tex]\(4x \times 2x^2 = 8x^3\)[/tex]
- [tex]\(4x \times 3x = 12x^2\)[/tex]
- [tex]\(4x \times (-4) = -16x\)[/tex]

- [tex]\(2 \times 2x^2 = 4x^2\)[/tex]
- [tex]\(2 \times 3x = 6x\)[/tex]
- [tex]\(2 \times (-4) = -8\)[/tex]

2. Combine all these terms:

[tex]\[
2x^4 + 3x^3 + 8x^3 + (-4x^2) + 12x^2 + 4x^2 + (-16x) + 6x + (-8)
\][/tex]

3. Simplify by combining like terms:

- [tex]\(2x^4\)[/tex] (no like terms for [tex]\(x^4\)[/tex])
- Combine [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- Combine [tex]\((-4x^2) + 12x^2 + 4x^2 = 12x^2\)[/tex]
- Combine [tex]\((-16x) + 6x = -10x\)[/tex]
- [tex]\(-8\)[/tex] (constant term)

So, after simplifying, we get the product:

[tex]\[
2x^4 + 11x^3 + 12x^2 - 10x - 8
\][/tex]

This matches option D.

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