This problem asks us to evaluate the limit:

$\lim_{x \to -1} \frac{x^2 - 19x - 20}{x + 1}.$

Step 1: Substitution Check

Substituting $x = -1$ into the denominator: $x + 1 = -1 + 1 = 0.$ The denominator becomes $0$, so direct substitution leads to an indeterminate form. We must simplify the expression.

Step 2: Factorize the Numerator

The numerator is: $x^2 - 19x - 20.$ Factoring it: $x^2 - 19x - 20 = (x - 20)(x + 1).$

Step 3: Simplify the Fraction

The fraction becomes: $\frac{(x - 20)(x + 1)}{x + 1}.$ For $x \neq -1$, we cancel $x + 1$, resulting in: $x - 20.$

Step 4: Evaluate the Limit

Now evaluate the simplified expression as $x \to -1$: $x - 20 \to -1 - 20 = -21.$

Final Answer:

$\lim_{x \to -1} \frac{x^2 - 19x - 20}{x + 1} = -21.$

Would you like further clarification or additional details on factoring or limits? Here are five related questions to deepen your understanding:

1. How can you identify and simplify indeterminate forms in limits?
2. What is the general process for factoring quadratic polynomials?
3. How do you handle limits involving higher-degree polynomials?
4. What is the difference between direct substitution and simplification in limit evaluation?
5. How does factoring help avoid undefined expressions in rational functions?

Tip: Always check for opportunities to factorize expressions when a direct substitution leads to indeterminate forms!