To solve the limit

$\lim_{x \to -2} \frac{x^2 + 8x + 12}{x^2 + 2x},$

we will first attempt to simplify the expression by factoring both the numerator and the denominator, if possible.

Step 1: Factor the numerator

The numerator is $x^2 + 8x + 12$. We factor this quadratic:

$x^2 + 8x + 12 = (x + 6)(x + 2).$

Step 2: Factor the denominator

The denominator is $x^2 + 2x$. We factor out the common factor $x$:

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

Step 3: Simplify the expression

Now, the original expression becomes:

$\frac{(x + 6)(x + 2)}{x(x + 2)}.$

We can cancel the common factor $(x + 2)$ from both the numerator and denominator (as long as $x \neq -2$):

$\frac{x + 6}{x}.$

Step 4: Evaluate the limit

Now, we evaluate the limit as $x \to -2$:

$\lim_{x \to -2} \frac{x + 6}{x} = \frac{-2 + 6}{-2} = \frac{4}{-2} = -2.$

Thus, the limit is:

$\boxed{-2}.$

Would you like more details or have any further questions?

Here are 5 related questions:

1. How do you handle limits that result in indeterminate forms like $\frac{0}{0}$?
2. What is the definition of a limit in calculus?
3. How do you factor quadratics quickly?
4. What are common techniques to evaluate limits without factoring?
5. How do you evaluate limits that approach infinity?

Tip: Always check if an expression can be simplified by factoring or canceling terms before directly substituting values in limits.