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The problem presented in your image asks to simplify the rational expression:

\[
\frac{8y^2 + 9y + 5}{y^2 - 7y - 8}
\]

Let's break this down by factoring both the numerator and the denominator.

### Step 1: Factor the numerator \( 8y^2 + 9y + 5 \)

We need to factor the quadratic expression. We look for two numbers that multiply to \( 8 \times 5 = 40 \) and add up to \( 9 \). These numbers are \( 5 \) and \( 8 \). We rewrite the middle term:

\[
8y^2 + 9y + 5 = 8y^2 + 5y + 4y + 5
\]

Now, factor by grouping:

\[
8y^2 + 5y + 4y + 5 = (8y^2 + 4y) + (5y + 5)
\]
\[
= 4y(2y + 1) + 5(2y + 1)
\]

So, the factored form of the numerator is:

\[
= (4y + 5)(2y + 1)
\]

### Step 2: Factor the denominator \( y^2 - 7y - 8 \)

We need to factor this quadratic expression. We look for two numbers that multiply to \( -8 \) and add up to \( -7 \). These numbers are \( -8 \) and \( 1 \). So we have:

\[
y^2 - 7y - 8 = (y - 8)(y + 1)
\]

### Step 3: Simplify the rational expression

Now, the expression becomes:

\[
\frac{(4y + 5)(2y + 1)}{(y - 8)(y + 1)}
\]

There are no common factors between the numerator and the denominator, so this is the simplified form of the rational expression.

\[
\boxed{\frac{(4y + 5)(2y + 1)}{(y - 8)(y + 1)}}
\]

Let me know if you would like further clarification or if you have any questions!

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Here are some related questions to think about:

1. How do you know when a quadratic expression is factorable?
2. What are the steps to factoring by grouping?
3. Can rational expressions always be simplified?
4. How do you find the domain of a rational function?
5. What happens if there is a common factor in both the numerator and denominator?

**Tip:** Always check if your rational expressions can be simplified by factoring, as this can make working with them much easier!

Simplify the rational expression 8y^2 + 9y + 5 / y^2 - 7y - 8

Learn how to simplify the rational expression 8y^2 + 9y + 5 / y^2 - 7y - 8 by factoring the numerator and denominator. This problem covers important algebra concepts like factoring and simplifying rational expressions. Suitable for students in grades 8-10.

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