Sure, let's simplify the given expression step-by-step:
[tex]\[
4 x\left(2 x^3-3 x^2+5 x\right)-3 x^2\left(2 x^2-1\right)
\][/tex]
First, we need to distribute [tex]\(4x\)[/tex] inside the first parentheses:
[tex]\[
4x(2x^3) + 4x(-3x^2) + 4x(5x)
\][/tex]
Calculating each term separately:
[tex]\[
= 4x \cdot 2x^3 + 4x \cdot (-3x^2) + 4x \cdot 5x
\][/tex]
[tex]\[
= 8x^4 - 12x^3 + 20x^2
\][/tex]
Next, we distribute [tex]\(-3x^2\)[/tex] inside the second parentheses:
[tex]\[
-3x^2(2x^2) + (-3x^2)(-1)
\][/tex]
Calculating each term separately:
[tex]\[
= -3x^2 \cdot 2x^2 + (-3x^2) \cdot (-1)
\][/tex]
[tex]\[
= -6x^4 + 3x^2
\][/tex]
Now, combine all the terms from both parts:
[tex]\[
8x^4 - 12x^3 + 20x^2 - 6x^4 + 3x^2
\][/tex]
Combine the like terms:
For the [tex]\(x^4\)[/tex] terms:
[tex]\[
8x^4 - 6x^4 = 2x^4
\][/tex]
For the [tex]\(x^3\)[/tex] terms:
[tex]\[
-12x^3
\][/tex]
(there are no other [tex]\(x^3\)[/tex] terms to combine with)
For the [tex]\(x^2\)[/tex] terms:
[tex]\[
20x^2 + 3x^2 = 23x^2
\][/tex]
The final simplified expression is:
[tex]\[
2x^4 - 12x^3 + 23x^2
\][/tex]
Thus, the correct answer is:
(B) [tex]\( 2x^4 - 12x^3 + 23x^2 \)[/tex]
