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Answer :
To multiply the polynomials
$$
(x^2 + 4x + 2)(2x^2 + 3x - 4),
$$
we follow these steps:
1. **Distribute Each Term:**
Multiply each term of the first polynomial by each term of the second polynomial.
- Multiply \(x^2\) by each term in \(2x^2 + 3x - 4\):
\[
\begin{aligned}
x^2 \times 2x^2 &= 2x^4,\\[1mm]
x^2 \times 3x &= 3x^3,\\[1mm]
x^2 \times (-4) &= -4x^2.
\end{aligned}
\]
- Multiply \(4x\) by each term in \(2x^2 + 3x - 4\):
\[
\begin{aligned}
4x \times 2x^2 &= 8x^3,\\[1mm]
4x \times 3x &= 12x^2,\\[1mm]
4x \times (-4) &= -16x.
\end{aligned}
\]
- Multiply \(2\) by each term in \(2x^2 + 3x - 4\):
\[
\begin{aligned}
2 \times 2x^2 &= 4x^2,\\[1mm]
2 \times 3x &= 6x,\\[1mm]
2 \times (-4) &= -8.
\end{aligned}
\]
2. **Combine Like Terms:**
Now, add all the results together:
- For the \(x^4\) term:
\[
2x^4.
\]
- For the \(x^3\) terms:
\[
3x^3 + 8x^3 = 11x^3.
\]
- For the \(x^2\) terms:
\[
-4x^2 + 12x^2 + 4x^2 = 12x^2.
\]
- For the \(x\) terms:
\[
-16x + 6x = -10x.
\]
- The constant term is:
\[
-8.
\]
3. **Write the Final Expression:**
Putting it all together, the product is:
$$
2x^4 + 11x^3 + 12x^2 - 10x - 8.
$$
Thus, the correct answer is:
$$\boxed{2x^4 + 11x^3 + 12x^2 - 10x - 8.}$$
$$
(x^2 + 4x + 2)(2x^2 + 3x - 4),
$$
we follow these steps:
1. **Distribute Each Term:**
Multiply each term of the first polynomial by each term of the second polynomial.
- Multiply \(x^2\) by each term in \(2x^2 + 3x - 4\):
\[
\begin{aligned}
x^2 \times 2x^2 &= 2x^4,\\[1mm]
x^2 \times 3x &= 3x^3,\\[1mm]
x^2 \times (-4) &= -4x^2.
\end{aligned}
\]
- Multiply \(4x\) by each term in \(2x^2 + 3x - 4\):
\[
\begin{aligned}
4x \times 2x^2 &= 8x^3,\\[1mm]
4x \times 3x &= 12x^2,\\[1mm]
4x \times (-4) &= -16x.
\end{aligned}
\]
- Multiply \(2\) by each term in \(2x^2 + 3x - 4\):
\[
\begin{aligned}
2 \times 2x^2 &= 4x^2,\\[1mm]
2 \times 3x &= 6x,\\[1mm]
2 \times (-4) &= -8.
\end{aligned}
\]
2. **Combine Like Terms:**
Now, add all the results together:
- For the \(x^4\) term:
\[
2x^4.
\]
- For the \(x^3\) terms:
\[
3x^3 + 8x^3 = 11x^3.
\]
- For the \(x^2\) terms:
\[
-4x^2 + 12x^2 + 4x^2 = 12x^2.
\]
- For the \(x\) terms:
\[
-16x + 6x = -10x.
\]
- The constant term is:
\[
-8.
\]
3. **Write the Final Expression:**
Putting it all together, the product is:
$$
2x^4 + 11x^3 + 12x^2 - 10x - 8.
$$
Thus, the correct answer is:
$$\boxed{2x^4 + 11x^3 + 12x^2 - 10x - 8.}$$
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