Please correct,if necessary, and simplify using long division: (-72-4x^2+8x^3-36x) devided by (x-3). My solution:

8x^2+20x+24
-------------------
8x^3-4x^2-36x-72
8x^3-24x^2
------------------
20x^2-36x-72
20x^2-60x
-----------------------
24x-72
24x-72
-------------------------

correct, the division was exact with no remainder.

the answer is 8x^2 + 20x + 24

synthetic division confirms it:

3 | 8 - 4 - 36 - 72
........ 24 .. 60 .. 72
.... 8..20 .. 24 ... 0

the 8, 20, 24 represents 8x^2 + 20x + 24
the 0 shows no remainder.

To perform long division, follow these steps:

1. Begin by dividing the first term of the dividend (8x^3) by the divisor (x-3). The result is the first term of the quotient, which is 8x^2.

2. Multiply the divisor (x-3) by the first term of the quotient (8x^2), giving you 8x^3 - 24x^2.

3. Subtract this product from the dividend: (-72 - 4x^2 + 8x^3 - 36x) - (8x^3 - 24x^2) = -4x^2 - 36x - 72.

4. Bring down the next term from the dividend, which is -4x^2. Now, you have -4x^2 - 36x - 72.

5. Divide the first term of this new expression (-4x^2) by the divisor (x-3), resulting in -4x.

6. Multiply the divisor (x-3) by the new term of the quotient (-4x), giving you -4x^2 + 12x.

7. Subtract this product from the previous expression: (-4x^2 - 36x - 72) - (-4x^2 + 12x) = -48x - 72.

8. Bring down the next term from the dividend, which is -48x. Now, you have -48x - 72.

9. Divide the first term of this new expression (-48x) by the divisor (x-3), resulting in -48.

10. Multiply the divisor (x-3) by the new term of the quotient (-48), giving you -48x + 144.

11. Subtract this product from the previous expression: (-48x - 72) - (-48x + 144) = -216.

12. Since there are no more terms to bring down, the division is complete.

The final quotient is 8x^2 + 20x + 24, with a remainder of -216.

So, the simplified expression after long division is 8x^2 + 20x + 24.