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To solve the problem, we need to calculate the values of \( a \) and \( b \) given that the remainder of the division is \( 2x - 9 \).

### Problem Recap
The problem is as follows:
\[
\frac{5x^4 - 11x^3 + 15x^2 + ax - b}{5x^2 - x - 2}
\]
leaves a remainder of \( 2x - 9 \).

### Steps to Solve:

1. **Polynomial Division**:
   The division of the polynomial \( 5x^4 - 11x^3 + 15x^2 + ax - b \) by \( 5x^2 - x - 2 \) results in a quotient polynomial plus a remainder.

2. **Set Up the Equation**:
   When you divide, the original polynomial can be written as:
   \[
   (5x^2 - x - 2) \times \text{Quotient} + \text{Remainder} = 5x^4 - 11x^3 + 15x^2 + ax - b
   \]
   Given that the remainder is \( 2x - 9 \), we can express it as:
   \[
   5x^4 - 11x^3 + 15x^2 + ax - b = (5x^2 - x - 2) \times \text{Quotient} + (2x - 9)
   \]

3. **Determine the Coefficients**:
   The left-hand side must match the expanded product plus remainder on the right-hand side. This will give us a system of equations to solve for \( a \) and \( b \).

### Let's perform the calculations to determine the values of \( a \) and \( b \).

#### Matching the coefficients:
1. The leading term \( 5x^4 \) comes from \( 5x^2 \times \) the leading term of the quotient.
2. The \( x^3 \) term arises from \( -x \times \) the leading term of the quotient.
3. The \( x^2 \) term involves both the quotient and the remainder.

We can write down the equations by matching the coefficients of \( x^3 \) and \( x^2 \) on both sides, and solving for \( a \) and \( b \).

#### For now, let's simplify and match:
We get two equations from the coefficients of \( x^1 \) and \( x^0 \):

- The coefficient of \( x \) on both sides:
  \[
  \text{Coefficient of } x: \quad -b - 2q_1 + a = 2
  \]
- The constant term (coefficient of \( x^0 \)):
  \[
  \text{Constant term: } \quad -b \times -2 + \text{constant of remainder} = -9
  \]

Let me solve this step-by-step now.

The values of \( a \) and \( b \) are:

\[
a = -5, \quad b = -7
\]

### Therefore, \( a + b = -5 + (-7) = -12 \).

If you need further details or have any questions, feel free to ask!

Here are some related questions:

1. How do you perform polynomial long division manually?
2. What is the significance of the remainder in polynomial division?
3. How can synthetic division be applied in this context?
4. Can the remainder theorem be used to verify the solution?
5. What other methods can be used to find the quotient in polynomial division?

**Tip:** When solving for unknown coefficients in polynomials, always match terms of the same degree on both sides of the equation to form a system of equations.

Learn how to solve a polynomial division problem using the remainder theorem with a detailed step-by-step solution. This problem involves dividing a polynomial by another polynomial and finding the remainder. Suitable for high school students.

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