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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 \( x = -1 \):

\[
\begin{array}{r|rrrrrr}
-1 & 1 & 3 & -5 & -15 & 4 & 12 \\
   &   & -1 & -2 & 7 & 8 & -12 \\
\hline
   & 1 & 2 & -7 & -8 & 12 & 0 \\
\end{array}
\]

The quotient is \( x^4 + 2x^3 - 7x^2 - 8x + 12 \) and the remainder is 0, confirming that the numerator is divisible by \( x + 1 \). Therefore, we have:

\[
\frac{x^5 + 3x^4 - 5x^3 - 15x^2 + 4x + 12}{x + 1} = x^4 + 2x^3 - 7x^2 - 8x + 12
\]

### Step 3: Substitute \( x = -1 \) into the simplified expression
Now, substitute \( x = -1 \) into the simplified polynomial:

\[
(-1)^4 + 2(-1)^3 - 7(-1)^2 - 8(-1) + 12 = 1 - 2 - 7 + 8 + 12 = 12
\]

### Final Answer:
\[
\lim_{x \to -1} \frac{x^5 + 3x^4 - 5x^3 - 15x^2 + 4x + 12}{x + 1} = 12
\]

Would you like further details on any step? 

Here are five related questions you might want to explore:
1. How does synthetic division work in detail?
2. What are indeterminate forms and how are they handled in calculus?
3. Can you always factor the numerator in a rational function when evaluating limits?
4. What happens if the remainder in synthetic division is not zero?
5. How is L'Hopital's Rule applied to solve indeterminate forms?

**Tip:** Always check for a common factor between the numerator and denominator before applying any advanced limit techniques!

Evaluate the limit: lim (x -> -1) (x^5 + 3x^4 - 5x^3 - 15x^2 + 4x + 12) / (x + 1)

Learn how to evaluate the limit of the polynomial fraction (x^5 + 3x^4 - 5x^3 - 15x^2 + 4x + 12) / (x + 1) as x approaches -1. This problem involves checking for direct substitution, using synthetic division, and factoring polynomials. Suitable for students in Grades 11-12.

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