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The problem you've provided asks to simplify the expression:

\[
\frac{16y^2 - 25}{12y^2 + 10 - 23y}
\]

### Step 1: Factor both the numerator and denominator.

- **Numerator: \(16y^2 - 25\)**

This is a difference of squares, which can be factored as:

\[
16y^2 - 25 = (4y - 5)(4y + 5)
\]

- **Denominator: \(12y^2 + 10 - 23y\)**

We can rearrange the terms for easier factoring:

\[
12y^2 - 23y + 10
\]

Now, factor this quadratic expression. Using the method of trial and error (or factoring by grouping):

We need factors of \(12 \times 10 = 120\) that add up to \(-23\). The correct pair is \(-15\) and \(-8\), so we rewrite the middle term:

\[
12y^2 - 15y - 8y + 10
\]

Now, factor by grouping:

\[
= 3y(4y - 5) - 2(4y - 5)
\]

Factor out the common term:

\[
= (4y - 5)(3y - 2)
\]

### Step 2: Simplify the expression.

The expression is now:

\[
\frac{(4y - 5)(4y + 5)}{(4y - 5)(3y - 2)}
\]

We can cancel out the common factor \((4y - 5)\) from the numerator and denominator:

\[
\frac{4y + 5}{3y - 2}
\]

### Final Answer:

The simplified expression is:

\[
\frac{4y + 5}{3y - 2}
\]

Would you like more details or have any other questions? 

Here are some related questions for further practice:
1. How do you factor a difference of squares?
2. What is factoring by grouping and when is it useful?
3. Can you explain how to factor a quadratic trinomial?
4. What happens if the numerator and denominator don't have common factors?
5. How can you verify that a fraction is fully simplified?

**Tip:** Always check for common factors in the numerator and denominator when simplifying rational expressions.

Simplify fully: (16y^2 - 25) / (12y^2 + 10 - 23y)

This page explains how to simplify the rational expression (16y^2 - 25) / (12y^2 + 10 - 23y) using factoring techniques. Learn how to factor a difference of squares and a quadratic expression step by step. Ideal for students in Grades 10-12.

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