Answer :
To find the product of [tex]\(2x^4(4x^2 + 3x + 1)\)[/tex], we need to distribute [tex]\(2x^4\)[/tex] to each term inside the parentheses. Let's do this step by step:
1. Multiply [tex]\(2x^4\)[/tex] by [tex]\(4x^2\)[/tex]:
[tex]\[
2x^4 \times 4x^2 = 8x^{4+2} = 8x^6
\][/tex]
The result of this multiplication is [tex]\(8x^6\)[/tex].
2. Multiply [tex]\(2x^4\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
2x^4 \times 3x = 6x^{4+1} = 6x^5
\][/tex]
The result of this multiplication is [tex]\(6x^5\)[/tex].
3. Multiply [tex]\(2x^4\)[/tex] by [tex]\(1\)[/tex]:
[tex]\[
2x^4 \times 1 = 2x^4
\][/tex]
The result of this multiplication is [tex]\(2x^4\)[/tex].
4. Combine all the terms:
Now, add all the terms together to form the final expanded expression:
[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]. This corresponds to the first option in the given choices.
1. Multiply [tex]\(2x^4\)[/tex] by [tex]\(4x^2\)[/tex]:
[tex]\[
2x^4 \times 4x^2 = 8x^{4+2} = 8x^6
\][/tex]
The result of this multiplication is [tex]\(8x^6\)[/tex].
2. Multiply [tex]\(2x^4\)[/tex] by [tex]\(3x\)[/tex]:
[tex]\[
2x^4 \times 3x = 6x^{4+1} = 6x^5
\][/tex]
The result of this multiplication is [tex]\(6x^5\)[/tex].
3. Multiply [tex]\(2x^4\)[/tex] by [tex]\(1\)[/tex]:
[tex]\[
2x^4 \times 1 = 2x^4
\][/tex]
The result of this multiplication is [tex]\(2x^4\)[/tex].
4. Combine all the terms:
Now, add all the terms together to form the final expanded expression:
[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]. This corresponds to the first option in the given choices.
