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The given expression to factor is:

\[
2a^2 - 11ab + 15b^2
\]

### Step-by-Step Solution:

We want to factor this quadratic expression of the form \(Ax^2 + Bxy + Cy^2\).

1. **Identify the coefficients:**
   - The coefficient of \(a^2\) is 2.
   - The coefficient of \(ab\) is -11.
   - The coefficient of \(b^2\) is 15.

2. **Split the middle term (coefficient of \(ab\)):**
   We need to find two numbers that multiply to \(2 \times 15 = 30\) and add up to -11. These numbers are -5 and -6.

3. **Rewrite the middle term using these numbers:**
   \[
   2a^2 - 6ab - 5ab + 15b^2
   \]

4. **Group the terms:**
   \[
   (2a^2 - 6ab) - (5ab - 15b^2)
   \]

5. **Factor each group:**
   \[
   2a(a - 3b) - 5b(a - 3b)
   \]

6. **Factor out the common binomial factor:**
   \[
   (2a - 5b)(a - 3b)
   \]

### Final Factorized Form:
\[
(2a - 5b)(a - 3b)
\]

Let me know if you need more details or explanations!

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Here are some related questions to deepen your understanding:

1. How do you determine which numbers to use when splitting the middle term?
2. Why is it important to factor the common binomial factor?
3. How can you verify if your factorization is correct?
4. What other methods can be used to factor quadratic expressions like this one?
5. How does the factorization change if the signs of the terms are different?

**Tip:** Always double-check your factorization by expanding the factored form to see if it gives you the original expression.

Learn how to factor the quadratic expression 2a^2 - 11ab + 15b^2 using a detailed step-by-step approach. This solution focuses on splitting the middle term and factoring the expression into two binomials. Perfect for students in Grades 8-10 studying algebra.

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