To solve the problem of rewriting [tex]\(\frac{2x^6 - 9x^5 + 4x^2 - 5}{x^3 - 5}\)[/tex] in the form [tex]\(q(x) + \frac{r(x)}{b(x)}\)[/tex], we need to perform polynomial division where:
- The dividend is [tex]\(2x^6 - 9x^5 + 4x^2 - 5\)[/tex]
- The divisor is [tex]\(x^3 - 5\)[/tex]
In the polynomial division, the quotient [tex]\(q(x)\)[/tex] is the result of the division, and [tex]\(r(x)\)[/tex] is the remainder.
Given the result from the polynomial division, the quotient [tex]\(q(x)\)[/tex] is:
[tex]\[2x^3 - 9x^2 + 10\][/tex]
Thus, the correct answer is:
[tex]\[q(x) = \boxed{2x^3 - 9x^2 + 10}\][/tex]
So, the answer is:
D. [tex]\(2 x^3 - 9 x^2 + 10\)[/tex]
