To perform the polynomial division of
[tex]\[
\frac{-9 x^4 + 4 x^2 + 15 - 14 x^3}{-x^2 - x + 2},
\][/tex]
we need to follow the steps of polynomial long division.
### Step-by-Step Solution
1. Arrange the Polynomials:
The dividend (numerator) should be ordered by descending powers of \( x \):
[tex]\[
-9x^4 - 14x^3 + 4x^2 + 0x + 15.
\][/tex]
The divisor (denominator) is:
[tex]\[
-x^2 - x + 2.
\][/tex]
2. Divide the Leading Terms:
Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{-9x^4}{-x^2} = 9x^2.
\][/tex]
3. Multiply and Subtract:
Multiply the entire divisor by this term and subtract from the dividend:
[tex]\[
\begin{aligned}
(-9x^4 - 14x^3 + 4x^2 + 0x + 15) - (9x^2 \cdot (-x^2 - x + 2)) & = (-9x^4 - 14x^3 + 4x^2 + 0x + 15) - (-9x^4 - 9x^3 + 18x^2) \\
& = 0x^4 - 5x^3 - 14x^2 + 0x + 15.
\end{aligned}
\][/tex]
4. Repeat the Process:
Now divide \(-5x^3\) by \(-x^2\):
[tex]\[
\frac{-5x^3}{-x^2} = 5x.
\][/tex]
Multiply and subtract again:
[tex]\[
\begin{aligned}
(-5x^3 - 14x^2 + 0x + 15) - (5x \cdot (-x^2 - x + 2)) & = (-5x^3 - 14x^2 + 0x + 15) - (-5x^3 - 5x^2 + 10x) \\
& = 0x^3 - 9x^2 - 10x + 15.
\end{aligned}
\][/tex]
5. Final Division and Remainder:
Now divide \(-9x^2\) by \(-x^2\):
[tex]\[
\frac{-9x^2}{-x^2} = 9.
\][/tex]
Multiply and subtract:
[tex]\[
\begin{aligned}
(-9x^2 - 10x + 15) - (9 \cdot (-x^2 - x + 2)) & = (-9x^2 - 10x + 15) - (-9x^2 - 9x + 18) \\
& = 0x^2 - x - 3.
\end{aligned}
\][/tex]
### Combining the Result
The quotient is:
[tex]\[
9x^2 + 5x + 9.
\][/tex]
The remainder is:
[tex]\[
-x - 3.
\][/tex]
So, the final answer in the requested form is:
[tex]\[
9x^2 + 5x + 9 + \frac{-x - 3}{-x^2 - x + 2}.
\][/tex]
