High School

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------------------------------------------------ Multiply:

\[
\begin{array}{r}
x^2 + 4x + 2 \\
\times \quad 2x^2 + 3x - 4 \\
\hline
\end{array}
\]

Choose the correct answer:

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 :

Sure! Let's walk through the process of multiplying the polynomials step by step to find the correct answer.

We want to multiply the two polynomials:

1. [tex]\( x^2 + 4x + 2 \)[/tex]
2. [tex]\( 2x^2 + 3x - 4 \)[/tex]

We'll multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.

### Steps:

1. Multiply the first term of the first polynomial [tex]\((x^2)\)[/tex] 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]

2. Multiply the second term of the first polynomial [tex]\((4x)\)[/tex] by each term in the second polynomial:

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

3. Multiply the third term of the first polynomial [tex]\((2)\)[/tex] by each term in the second polynomial:

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

### Combine all the results:

Now, let's add together all the terms obtained from each step:

- For [tex]\(x^4\)[/tex] terms: [tex]\(2x^4\)[/tex]
- For [tex]\(x^3\)[/tex] terms: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(-16x + 6x = -10x\)[/tex]
- Constant term: [tex]\(-8\)[/tex]

Final polynomial:

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

So, the correct answer is D. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].