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To factor the quadratic expression \( 4x^2 - 23xy + 15y^2 \), we will use the method of factoring by grouping or using the ac-method. Let's proceed with the steps:

1. **Rewrite the quadratic equation**: 
   The given quadratic is:
   \[
   4x^2 - 23xy + 15y^2
   \]
   Here, the coefficients are:
   - \( a = 4 \)
   - \( b = -23 \)
   - \( c = 15 \)

2. **Multiply \( a \) and \( c \)**:
   Calculate the product \( ac = 4 \cdot 15 = 60 \).

3. **Find two numbers that multiply to \( ac \) and add up to \( b \)**:
   We need to find two numbers whose product is \( 60 \) and whose sum is \( -23 \). These numbers are \( -20 \) and \( -3 \).

4. **Rewrite the middle term using these two numbers**:
   Rewrite the expression as:
   \[
   4x^2 - 20xy - 3xy + 15y^2
   \]

5. **Group the terms**:
   Group the terms into two pairs:
   \[
   (4x^2 - 20xy) - (3xy - 15y^2)
   \]

6. **Factor out the common factors in each group**:
   \[
   4x(x - 5y) - 3y(x - 5y)
   \]

7. **Factor out the common binomial**:
   \[
   (4x - 3y)(x - 5y)
   \]

Therefore, the factored form of the expression \( 4x^2 - 23xy + 15y^2 \) is:
\[
(4x - 3y)(x - 5y)
\]

Would you like more details about this factorization process or have any questions?

Here are five related questions to expand your understanding:
1. How do you identify the correct pair of numbers in the ac-method for factoring?
2. What is the difference between factoring by grouping and using the quadratic formula?
3. Can all quadratic expressions be factored using integers?
4. What are the applications of factoring quadratic expressions in real-world problems?
5. How would the factorization process change if one of the coefficients was a fraction?

**Tip:** When factoring quadratics, always check if there's a greatest common factor (GCF) that can be factored out first to simplify the expression.

Factor the quadratic expression 4x^2 - 23xy + 15y^2.

Learn how to factor the quadratic expression 4x^2 - 23xy + 15y^2 using the ac-method. This problem involves breaking down the middle term and grouping, suitable for high school students in Grades 9-12.

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