The given expression is:

\[
m^2 - 8mn + 15n^2
\]

This is a quadratic expression in terms of \( m \) and \( n \). To factor it, we look for two binomials whose product results in this expression.

### Step-by-step factorization:
1. We need two numbers that multiply to \( 15n^2 \) and add up to \( -8n \). The factors of \( 15n^2 \) are:
   - \( (3n)(5n) \), because \( 3n \times 5n = 15n^2 \).
   
2. These factors also add up to give \( -8n \), as:
   - \( -3n - 5n = -8n \).

Therefore, the factorized form is:
\[
(m - 3n)(m - 5n)
\]

So the factorization of the given expression \( m^2 - 8mn + 15n^2 \) is:
\[
(m - 3n)(m - 5n)
\]

### Do you want more details, or have any questions?

Here are 5 related questions to expand your understanding:
1. How can we apply the same method to factor other quadratic expressions?
2. What is the general form of a quadratic expression in two variables?
3. How would you expand \( (m - 3n)(m - 5n) \) back to verify the factorization?
4. What are the key differences between factoring quadratics in one variable versus two variables?
5. How can factoring quadratics help solve algebraic equations?

**Tip:** When factoring quadratics, always check by multiplying the factors back to ensure the result matches the original expression!

Discover how to factor the quadratic expression m^2 - 8mn + 15n^2 into (m - 3n)(m - 5n) using algebraic methods. This problem is ideal for students learning to factor quadratic expressions with two variables. Suitable for Grades 9-10.

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