High School

Multiply the following expressions:

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

Choose the correct product:

A. \([2x^4 + 23x^2 - 10x - 8]\)
B. \([3x^2 + 7x - 2]\)
C. \([2x^4 + 11x^3 + 12x^2 - 10x - 8]\)
D. \([2x^4 + 12x^2 - 8]\)

Answer :

Sure! Let's multiply the two polynomials step by step:

We are given:
[tex]\[ (x^2 + 4x + 2) \times (2x^2 + 3x - 4) \][/tex]

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

### Step-by-step calculation:

1. Multiply each term in [tex]\(x^2 + 4x + 2\)[/tex] by [tex]\(2x^2\)[/tex]:
- [tex]\(x^2 \times 2x^2 = 2x^4\)[/tex]
- [tex]\(4x \times 2x^2 = 8x^3\)[/tex]
- [tex]\(2 \times 2x^2 = 4x^2\)[/tex]

2. Multiply each term in [tex]\(x^2 + 4x + 2\)[/tex] by [tex]\(3x\)[/tex]:
- [tex]\(x^2 \times 3x = 3x^3\)[/tex]
- [tex]\(4x \times 3x = 12x^2\)[/tex]
- [tex]\(2 \times 3x = 6x\)[/tex]

3. Multiply each term in [tex]\(x^2 + 4x + 2\)[/tex] by [tex]\(-4\)[/tex]:
- [tex]\(x^2 \times -4 = -4x^2\)[/tex]
- [tex]\(4x \times -4 = -16x\)[/tex]
- [tex]\(2 \times -4 = -8\)[/tex]

### Adding all these results together:

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

Therefore, the expanded form of the polynomial is:
[tex]\[ 2x^4 + 11x^3 + 12x^2 - 10x - 8 \][/tex]

The correct answer is C. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].

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