Question

Simplify the rational expression. If the rational expression cannot be​ simplified, so state.
StartFraction x plus 3 Over x squared minus 4 x minus 21 EndFraction
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Part 1
Select the correct choice below and fill in any answer boxes in your choice.
A.StartFraction x plus 3 Over x squared minus 4 x minus 21 EndFraction equals enter your response here
StartFraction x plus 3 Over x squared minus 4 x minus 21 EndFraction
equals
  
enter your response here
B.
The expression cannot be simplified.

Answer 1

To simplify the rational expression
\[
\frac{x + 3}{x^2 - 4x - 21},
\]
we first need to factor the denominator.

The expression in the denominator is a quadratic expression:
\[
x^2 - 4x - 21.
\]

We can factor this expression by finding two numbers that multiply to \(-21\) (the constant term) and add to \(-4\) (the coefficient of the linear term). The numbers \(-7\) and \(3\) work because:
- \(-7 \cdot 3 = -21\)
- \(-7 + 3 = -4\)

Thus, we can factor the denominator as:
\[
x^2 - 4x - 21 = (x - 7)(x + 3).
\]

Substituting this back into the rational expression, we get:
\[
\frac{x + 3}{(x - 7)(x + 3)}.
\]

Now, we can cancel the common factor of \((x + 3)\) from the numerator and denominator (as long as \(x \neq -3\)):
\[
\frac{1}{x - 7}, \quad \text{for } x \neq -3.
\]

Thus, the simplified expression is:
\[
\frac{1}{x - 7}.
\]

### Final Answer:
A.
\[
\frac{x + 3}{x^2 - 4x - 21} = \frac{1}{x - 7}, \quad x \neq -3.
\]