To find the quotient when [tex]\(\left(16 x^4 + 40 x^3 - 24 x^2\right)\)[/tex] is divided by [tex]\(8 x^2\)[/tex], we can proceed with polynomial division by performing each term one by one. Here's how you can do it step-by-step:
1. Divide the leading terms of the numerator by the leading term of the divisor:
[tex]\[
\frac{16 x^4}{8 x^2} = 2 x^2
\][/tex]
2. Multiply the entire divisor by this result:
[tex]\[
(2 x^2) \cdot (8 x^2) = 16 x^4
\][/tex]
3. Subtract this result from the original polynomial:
[tex]\[
(16 x^4 + 40 x^3 - 24 x^2) - (16 x^4) = 40 x^3 - 24 x^2
\][/tex]
4. Repeat the process with the new polynomial [tex]\(40 x^3 - 24 x^2\)[/tex]:
[tex]\[
\frac{40 x^3}{8 x^2} = 5 x
\][/tex]
5. Multiply the entire divisor by this new result:
[tex]\[
(5 x) \cdot (8 x^2) = 40 x^3
\][/tex]
6. Subtract this result from the current polynomial:
[tex]\[
(40 x^3 - 24 x^2) - (40 x^3) = -24 x^2
\][/tex]
7. Repeat the process with the new term [tex]\(-24 x^2\)[/tex]:
[tex]\[
\frac{-24 x^2}{8 x^2} = -3
\][/tex]
8. Multiply the entire divisor by this new result:
[tex]\[
(-3) \cdot (8 x^2) = -24 x^2
\][/tex]
9. Subtract this result from the current polynomial:
[tex]\[
(-24 x^2) - (-24 x^2) = 0
\][/tex]
So, the quotient is given by the combination of the results from each step:
[tex]\[
\boxed{2 x^2 + 5 x - 3}
\][/tex]
Therefore, the correct answer is:
D. [tex]\(2 x^2 + 5 x - 3\)[/tex].
