The expression you've provided is:

$\frac{x^2 - 5x - 14}{x^2 + 4x + 4}$

Let's first simplify the expression by factoring both the numerator and the denominator, if possible.

Step 1: Factor the denominator $x^2 + 4x + 4$.

This is a perfect square trinomial:

$x^2 + 4x + 4 = (x + 2)(x + 2) = (x + 2)^2$

Step 2: Factor the numerator $x^2 - 5x - 14$.

We need to find two numbers that multiply to $-14$ and add to $-5$. The numbers $-7$ and $2$ satisfy this condition:

$x^2 - 5x - 14 = (x - 7)(x + 2)$

Step 3: Simplify the expression.

Now, rewrite the entire expression:

$\frac{(x - 7)(x + 2)}{(x + 2)^2}$

We can cancel one factor of $x + 2$ from the numerator and denominator:

$\frac{x - 7}{x + 2} \quad \text{(for } x \neq -2\text{, since the denominator cannot be zero)}$

Thus, the simplified expression is:

$\frac{x - 7}{x + 2}, \quad x \neq -2$

Let me know if you have any questions or if you'd like further details.

Related Questions:

1. How do you find common factors in algebraic expressions?
2. What is a perfect square trinomial and how do you recognize it?
3. Can you explain the restrictions on variables in rational expressions?
4. How do you handle polynomials that are not easily factorable?
5. How do you multiply and divide rational expressions?

Tip:

Always check for values that make the denominator zero when simplifying rational expressions to avoid undefined expressions.