How do you divide #(-2x^3+6x^2+9x+15)/(x+1) #?

To divide (-2x^3+6x^2+9x+15) by (x+1), you can use long division or synthetic division. Here is the step-by-step process using long division:

1. Divide the first term of the numerator (-2x^3) by the first term of the denominator (x). This gives -2x^2.
2. Multiply the entire denominator (x+1) by -2x^2, which gives -2x^3-2x^2.
3. Subtract this result (-2x^3-2x^2) from the numerator (-2x^3+6x^2+9x+15). This gives 8x^2+9x+15.
4. Bring down the next term from the numerator, which is 8x^2. Now you have 8x^2+9x+15.
5. Divide the first term of this new expression (8x^2) by the first term of the denominator (x). This gives 8x.
6. Multiply the entire denominator (x+1) by 8x, which gives 8x^2+8x.
7. Subtract this result (8x^2+8x) from the previous expression (8x^2+9x+15). This gives x+15.
8. Bring down the next term from the numerator, which is x. Now you have x+15.
9. Divide the first term of this new expression (x) by the first term of the denominator (x). This gives 1.
10. Multiply the entire denominator (x+1) by 1, which gives x+1.
11. Subtract this result (x+1) from the previous expression (x+15). This gives 14.
12. There are no more terms left in the numerator, so the division is complete.

The quotient is -2x^2 + 8x + 1, and the remainder is 14.