To find the correct product of [tex]\( 2x^4(4x^2 + 3x + 1) \)[/tex], let us follow a detailed, step-by-step approach.
1. Rewrite the expression:
We need to multiply the polynomial [tex]\( 4x^2 + 3x + 1 \)[/tex] by [tex]\( 2x^4 \)[/tex].
[tex]\[
2x^4 (4x^2 + 3x + 1)
\][/tex]
2. Distribute the [tex]\( 2x^4 \)[/tex] across each term inside the parentheses:
[tex]\[
2x^4 \cdot 4x^2 + 2x^4 \cdot 3x + 2x^4 \cdot 1
\][/tex]
3. Multiply each term individually:
[tex]\[
= 2x^4 \cdot 4x^2 + 2x^4 \cdot 3x + 2x^4 \cdot 1
\][/tex]
[tex]\[
= 2 \cdot 4 \cdot x^{4+2} + 2 \cdot 3 \cdot x^{4+1} + 2 \cdot 1 \cdot x^4
\][/tex]
[tex]\[
= 8x^6 + 6x^5 + 2x^4
\][/tex]
So the product of [tex]\( 2x^4 (4x^2 + 3x + 1) \)[/tex] is [tex]\( 8x^6 + 6x^5 + 2x^4 \)[/tex].
From the given choices, the correct answer is:
[tex]\[
8x^6 + 6x^5 + 2x^4
\][/tex]
The other options do not match this result.
