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------------------------------------------------ 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. $[2x^4 + 12x^2 - 8]$

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].