We are tasked with evaluating the limit:

$\lim_{x \to -1} \frac{x^5 + 3x^4 - 5x^3 - 15x^2 + 4x + 12}{x + 1}$

Step 1: Check for direct substitution

We first check if the limit can be evaluated directly by substituting $x = -1$ into the expression.

For the numerator: $(-1)^5 + 3(-1)^4 - 5(-1)^3 - 15(-1)^2 + 4(-1) + 12 = -1 + 3 + 5 - 15 - 4 + 12 = 0$

For the denominator: $x + 1 = -1 + 1 = 0$

Since both the numerator and denominator evaluate to 0, we have an indeterminate form $\frac{0}{0}$, and we need to simplify the expression by factoring.

Step 2: Factor the numerator

We attempt to factor the numerator $x^5 + 3x^4 - 5x^3 - 15x^2 + 4x + 12$. Since we are dividing by $x + 1$, we can use synthetic division or polynomial division to factor the numerator.

Using synthetic division with $x = -1$: