https://file.aiquickdraw.com/mathbot/file/172537129898430hmj8cw.jpg

To simplify the expression given in the image:

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

we need to factor both the numerator and the denominator where possible.

### Step 1: Factor the Numerator and Denominator

The expression is:

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

1. **Numerator:** \(y^2 + 5x + 4\)

   We need to factor this quadratic expression. To factor a quadratic in the form \(ax^2 + bx + c\), we look for two numbers that multiply to \(ac\) and add up to \(b\). 

2. **Denominator:** \(y^2 - 3x - 4\)

   Similarly, we will factor this quadratic expression by finding two numbers that multiply to \(ac\) and add up to \(b\).

Let me proceed with factoring each of these.Let's factor the numerator and the denominator step-by-step:

### Step 1: Factor the Numerator: \(y^2 + 5x + 4\)

For \(y^2 + 5x + 4\), we need two numbers that multiply to \(1 \cdot 4 = 4\) and add up to \(5\). Those numbers are \(4\) and \(1\). Therefore:

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

### Step 2: Factor the Denominator: \(y^2 - 3x - 4\)

For \(y^2 - 3x - 4\), we need two numbers that multiply to \(1 \cdot (-4) = -4\) and add up to \(-3\). Those numbers are \(-4\) and \(1\). Therefore:

\[
y^2 - 3x - 4 = (y - 4)(y + 1)
\]

### Step 3: Simplify the Expression

Now, the expression becomes:

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

We can cancel out the common factor \((y + 1)\) in both the numerator and the denominator:

\[
\frac{y + 4}{y - 4}
\]

### Simplified Expression:

The simplified form of the given expression is:

\[
\frac{y + 4}{y - 4}
\]

### Note:

- This simplification is valid as long as \(y \neq -1\) (to avoid division by zero in the original expression).

Would you like more details on any of these steps or have any questions? Here are some related questions that you might find helpful:

1. How do you factor a quadratic expression in general?
2. What are the cases when you cannot simplify a rational expression?
3. How does canceling common factors work in algebra?
4. What happens if \(y = -1\) in the original expression?
5. How can we apply this technique to more complex expressions?

**Tip:** Always check for any restrictions on the variables when simplifying algebraic expressions to avoid dividing by zero.

Simplify the expression (y^2 + 5x + 4) / (y^2 - 3x - 4).

This problem demonstrates how to simplify the expression (y^2 + 5x + 4) / (y^2 - 3x - 4) by factoring both the numerator and the denominator. This example involves key algebraic techniques such as factoring quadratic expressions and simplifying rational expressions. Suitable for students in Grades 8-10.

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