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We are tasked with dividing the polynomial:

\[
\frac{-10x^3 + 15x^2 + 15x - 8}{5x + 5}
\]

### Steps:

1. **Simplify the divisor:**
   \[
   5x + 5 = 5(x + 1)
   \]
   So, we are essentially dividing by \(5(x + 1)\).

2. **Divide each term of the numerator by 5 first:**
   \[
   \frac{-10x^3 + 15x^2 + 15x - 8}{5} = -2x^3 + 3x^2 + 3x - \frac{8}{5}
   \]

3. **Now perform polynomial long division** of \(-2x^3 + 3x^2 + 3x - \frac{8}{5}\) by \(x + 1\):

    - Divide the first term: \(\frac{-2x^3}{x} = -2x^2\)
      Multiply: \((-2x^2)(x + 1) = -2x^3 - 2x^2\)
      Subtract: \((-2x^3 + 3x^2) - (-2x^3 - 2x^2) = 5x^2\)

    - Divide the next term: \(\frac{5x^2}{x} = 5x\)
      Multiply: \((5x)(x + 1) = 5x^2 + 5x\)
      Subtract: \((5x^2 + 3x) - (5x^2 + 5x) = -2x\)

    - Divide the next term: \(\frac{-2x}{x} = -2\)
      Multiply: \((-2)(x + 1) = -2x - 2\)
      Subtract: \((-2x - \frac{8}{5}) - (-2x - 2) = -\frac{8}{5} + 2 = \frac{2}{5}\)

4. **Final answer in the quotient + remainder/divisor form**:
   \[
   -2x^2 + 5x - 2 + \frac{\frac{2}{5}}{x + 1} = -2x^2 + 5x - 2 + \frac{2}{5(x + 1)}
   \]

So, the result is:

\[
\boxed{-2x^2 + 5x - 2 + \frac{2}{5(x + 1)}}
\]

Would you like further details on any step? Here are 5 related questions:

1. What are the steps for simplifying complex fractions?
2. How do you perform polynomial long division in more detail?
3. Can you explain why we simplify the divisor before dividing?
4. What happens when the remainder is zero in polynomial division?
5. How can synthetic division be used for similar problems?

**Tip:** Always check if the divisor can be factored out for easier division.

Divide the polynomial (-10x^3 + 15x^2 + 15x - 8) by (5x + 5) and express the answer in the form quotient + remainder/divisor if necessary.

Learn how to divide the polynomial -10x^3 + 15x^2 + 15x - 8 by 5x + 5. This problem uses the polynomial long division method and the division algorithm for polynomials. Suitable for students in Grades 9-11.

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