https://file.aiquickdraw.com/mathbot/file/1726272454069z5ocg5oa.jpg

The question in the image asks to simplify the following rational expression:

\[
\frac{x^2 - 5x - 14}{x^2 + 5x + 6}
\]

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

#### Numerator: \( x^2 - 5x - 14 \)
We need to factor this quadratic expression. We look for two numbers that multiply to \(-14\) (the constant term) and add to \(-5\) (the coefficient of \(x\)).

The two numbers are \(-7\) and \(2\). So, we can factor the numerator as:
\[
x^2 - 5x - 14 = (x - 7)(x + 2)
\]

#### Denominator: \( x^2 + 5x + 6 \)
Now, we factor the denominator by finding two numbers that multiply to \(6\) (the constant term) and add to \(5\) (the coefficient of \(x\)).

The two numbers are \(2\) and \(3\). So, the denominator factors as:
\[
x^2 + 5x + 6 = (x + 2)(x + 3)
\]

### Step 2: Simplify the rational expression
Now that we have factored both the numerator and the denominator, the expression becomes:

\[
\frac{(x - 7)(x + 2)}{(x + 2)(x + 3)}
\]

We can cancel the common factor of \( (x + 2) \) from the numerator and the denominator:

\[
\frac{x - 7}{x + 3}
\]

### Step 3: Determine the restrictions
The original expression has restrictions based on the denominator. The denominator cannot be zero, so we need to solve:

\[
x + 2 \neq 0 \quad \text{and} \quad x + 3 \neq 0
\]
\[
x \neq -2 \quad \text{and} \quad x \neq -3
\]

### Final Answer:
The simplified expression is:

\[
\frac{x - 7}{x + 3}, \quad x \neq -3, -2
\]

The correct choice in the image is the last one:
\[
\frac{x - 7}{x + 3}, \quad x \neq -3, -2
\]

Would you like further details on this solution or have any other questions?

Here are 5 related questions you might find useful:
1. How do you factor trinomials when the leading coefficient is not 1?
2. What happens if both the numerator and denominator are perfect squares?
3. Can you always cancel terms in a rational expression?
4. How do restrictions in a rational expression affect the graph of the function?
5. What are asymptotes, and how do they relate to restrictions in rational functions?

**Tip:** Always factor both the numerator and denominator before canceling terms in a rational expression.

Simplify the rational expression: (x^2 - 5x - 14) / (x^2 + 5x + 6).

Learn how to simplify the rational expression (x^2 - 5x - 14) / (x^2 + 5x + 6) through factoring and canceling common factors. This problem covers key algebraic concepts, including factoring quadratics and simplifying rational expressions. Suitable for students in Grades 8-10.

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