Question

Simplify the rational expression. If the rational expression cannot be​ simplified, so state.
StartFraction x plus 4 Over x squared minus x minus 20 EndFraction
Question content area bottom
Part 1
Select the correct choice below and fill in any answer boxes in your choice.
A.StartFraction x plus 4 Over x squared minus x minus 20 EndFraction equals enter your response here
StartFraction x plus 4 Over x squared minus x minus 20 EndFraction
equals
  
enter your response here
B.
The expression cannot be simplified.

Answer 1

To simplify the rational expression \( \frac{x + 4}{x^2 - x - 20} \), we first need to factor the denominator.

1. **Factor the denominator**:
We need to factor \( x^2 - x - 20 \).
We are looking for two numbers that multiply to \(-20\) and add to \(-1\). These numbers are \(-5\) and \(4\).
Therefore, we can factor the expression as follows:
\[
x^2 - x - 20 = (x - 5)(x + 4)
\]

2. **Rewrite the expression**:
Now substitute the factored form of the denominator into the rational expression:
\[
\frac{x + 4}{x^2 - x - 20} = \frac{x + 4}{(x - 5)(x + 4)}
\]

3. **Cancel common factors**:
The \( x + 4 \) in the numerator and denominator can be canceled out:
\[
\frac{x + 4}{(x - 5)(x + 4)} = \frac{1}{x - 5} \quad \text{(for } x \neq -4\text{)}
\]

Thus, the simplified expression is:

\[
\frac{1}{x - 5}
\]

In conclusion, the expression can be simplified to \( \frac{1}{x - 5} \) as long as \( x \neq -4 \) and \( x \neq 5 \) (to avoid division by zero).

The final answer for your selection is:

**A.**
\[
\frac{x + 4}{x^2 - x - 20} = \frac{1}{x - 5}
\]