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Find the product of [tex]2x^4(2x^2+3x+4)[/tex].

A. [tex]2x^8 + 3x^4 + 4x^4[/tex]

B. [tex]4x^6 + 6x^5 + 8x^4[/tex]

C. [tex]4x^4 + 3x^5 + 2x^6[/tex]

D. [tex]3x^6 + 4x^5 + 5x^4[/tex]

Answer :

To find the product of [tex]\(2x^4(2x^2 + 3x + 4)\)[/tex], we'll multiply each term inside the parentheses by [tex]\(2x^4\)[/tex] and combine the like terms. Let's go step-by-step:

1. Multiply [tex]\(2x^4\)[/tex] by each term in the expression:

- Multiply [tex]\(2x^4\)[/tex] by the first term [tex]\(2x^2\)[/tex]:
[tex]\[
2x^4 \times 2x^2 = 4x^{4+2} = 4x^6
\][/tex]

- Multiply [tex]\(2x^4\)[/tex] by the second term [tex]\(3x\)[/tex]:
[tex]\[
2x^4 \times 3x = 6x^{4+1} = 6x^5
\][/tex]

- Multiply [tex]\(2x^4\)[/tex] by the third term [tex]\(4\)[/tex]:
[tex]\[
2x^4 \times 4 = 8x^4
\][/tex]

2. Combine all these products to get the final result:

[tex]\[
4x^6 + 6x^5 + 8x^4
\][/tex]

So, the product of [tex]\(2x^4(2x^2 + 3x + 4)\)[/tex] is [tex]\(4x^6 + 6x^5 + 8x^4\)[/tex]. This matches one of the given answer choices.

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