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]
