To divide the polynomial [tex]\(-20x^3 - 9x^2 + 36x + 20\)[/tex] by [tex]\(-4x - 5\)[/tex] using long division, we will follow these steps:
1. Set up the division: Write the polynomial [tex]\(-20x^3 - 9x^2 + 36x + 20\)[/tex] (the dividend) under the division symbol and [tex]\(-4x - 5\)[/tex] (the divisor) outside it.
2. Divide the leading terms:
- Divide the leading term of the dividend, [tex]\(-20x^3\)[/tex], by the leading term of the divisor, [tex]\(-4x\)[/tex].
- [tex]\(-20x^3\)[/tex] divided by [tex]\(-4x\)[/tex] is [tex]\(5x^2\)[/tex].
3. Multiply and subtract:
- Multiply the divisor, [tex]\(-4x - 5\)[/tex], by the result from step 2, [tex]\(5x^2\)[/tex]:
[tex]\((-4x - 5) \cdot 5x^2 = -20x^3 - 25x^2\)[/tex].
- Subtract this result from the original polynomial:
[tex]\((-20x^3 - 9x^2 + 36x + 20) - (-20x^3 - 25x^2)\)[/tex].
- This gives [tex]\(16x^2 + 36x + 20\)[/tex].
4. Repeat the process:
- Divide the new leading term of the updated dividend, [tex]\(16x^2\)[/tex], by the leading term of the divisor, [tex]\(-4x\)[/tex].
- [tex]\(16x^2\)[/tex] divided by [tex]\(-4x\)[/tex] is [tex]\(-4x\)[/tex].
- Multiply the divisor by [tex]\(-4x\)[/tex]:
[tex]\((-4x - 5) \cdot -4x = 16x^2 + 20x\)[/tex].
- Subtract this from the current polynomial:
[tex]\((16x^2 + 36x + 20) - (16x^2 + 20x)\)[/tex].
- This gives [tex]\(16x + 20\)[/tex].
5. Divide the remaining term:
- Divide the new leading term [tex]\(16x\)[/tex] by the leading term of the divisor [tex]\(-4x\)[/tex].
- [tex]\(16x\)[/tex] divided by [tex]\(-4x\)[/tex] is [tex]\(-4\)[/tex].
- Multiply the divisor by [tex]\(-4\)[/tex]:
[tex]\((-4x - 5) \cdot -4 = 16x + 20\)[/tex].
- Subtract this from the remaining polynomial:
[tex]\((16x + 20) - (16x + 20)\)[/tex].
- This yields a remainder of 0.
Putting all the steps together, the quotient is [tex]\(5x^2 - 4x - 4\)[/tex] and the remainder is 0.
Thus, the correct answer is:
[tex]\[
\boxed{5x^2 - 4x - 4}
\][/tex]
