poopey
Answered

What is the product?

[tex]\[
\left(x^4\right)\left(3x^3 - 2\right)\left(4x^2 + 5x\right)
\][/tex]

A. \(12x^9 + 15x^8 - 8x^6 - 10x^5\)

B. \(12x^{24} + 15x^{12} - 8x^8 - 10x^4\)

C. \(12x^9 - 10x^5\)

D. [tex]\(12x^{24} - 10x^4\)[/tex]



Answer :

To find the product \(\left(x^4\right)\left(3 x^3 - 2\right)\left(4 x^2 + 5 x\right)\), we need to distribute each term carefully through multiplication.

Step 1: Multiply \(x^4\) and \((3 x^3 - 2)\) first.

[tex]\[ x^4 \cdot (3 x^3 - 2) = x^4 \cdot 3 x^3 - x^4 \cdot 2 \][/tex]

[tex]\[ = 3 x^7 - 2 x^4 \][/tex]

Step 2: Multiply the result with \((4 x^2 + 5 x)\).

Now we distribute \((4 x^2 + 5 x)\) across each term in the expression \(3 x^7 - 2 x^4\):

[tex]\[ (3 x^7 - 2 x^4) \cdot (4 x^2 + 5 x) \][/tex]

1. Distribute \(3 x^7\):

[tex]\[ 3 x^7 \cdot 4 x^2 = 12 x^9 \][/tex]

[tex]\[ 3 x^7 \cdot 5 x = 15 x^8 \][/tex]

2. Distribute \(-2 x^4\):

[tex]\[ -2 x^4 \cdot 4 x^2 = -8 x^6 \][/tex]

[tex]\[ -2 x^4 \cdot 5 x = -10 x^5 \][/tex]

Step 3: Combine all terms together:

[tex]\[ 12 x^9 + 15 x^8 - 8 x^6 - 10 x^5 \][/tex]

Thus, the product \(\left(x^4\right)\left(3 x^3 - 2\right)\left(4 x^2 + 5 x\right)\) simplifies to:

[tex]\[ 12 x^9 + 15 x^8 - 8 x^6 - 10 x^5 \][/tex]

Therefore, the correct answer is:

[tex]\[ 12 x^9 + 15 x^8 - 8 x^6 - 10 x^5 \][/tex]