How do you divide #(2x^3 + 13x + 15)/(3x + 12)#?

To divide (2x^3 + 13x + 15) by (3x + 12), you can use long division. Here are the steps:

1. Divide the first term of the numerator (2x^3) by the first term of the denominator (3x). The result is (2/3)x^2.

2. Multiply the entire denominator (3x + 12) by the result from step 1, which is (2/3)x^2. This gives you (2/3)x^2 * (3x + 12) = (2/3)x^2 * 3x + (2/3)x^2 * 12 = 2x^3 + 8x^2.

3. Subtract the result from step 2 from the numerator (2x^3 + 13x + 15). This gives you (2x^3 + 13x + 15) - (2x^3 + 8x^2) = 13x + 15 - 8x^2.

4. Bring down the next term from the numerator, which is 13x. The expression becomes 13x + 15 - 8x^2.

5. Divide the first term of the new expression (13x) by the first term of the denominator (3x). The result is (13/3).

6. Multiply the entire denominator (3x + 12) by the result from step 5, which is (13/3). This gives you (13/3) * (3x + 12) = (13/3) * 3x + (13/3) * 12 = 13x + 52.

7. Subtract the result from step 6 from the new expression (13x + 15 - 8x^2). This gives you (13x + 15 - 8x^2) - (13x + 52) = -8x^2 - 37.

8. Bring down the next term from the numerator, which is -37. The expression becomes -8x^2 - 37.

9. Divide the first term of the new expression (-8x^2) by the first term of the denominator (3x). The result is (-8/3)x.

10. Multiply the entire denominator (3x + 12) by the result from step 9, which is (-8/3)x. This gives you (-8/3)x * (3x + 12) = (-8/3)x * 3x + (-8/3)x * 12 = -8x^2 - 32x.

11. Subtract the result from step 10 from the new expression (-8x^2 - 37). This gives you (-8x^2 - 37) - (-8x^2 - 32x) = -37 - (-32x) = -37 + 32x.

The final result of the division is (-8/3)x + 32 + (-37 + 32x)/(3x + 12).