Simplify the expression and state the restrictions on the variable. \( \frac{r^{2}-9}{r^{2}-4 r-21} \) Part 1 of 2 Simplify. \( \frac{r^{2}-9}{r^{2}-4 r-21}=\square \)

To simplify the expression \( \frac{r^{2}-9}{r^{2}-4 r-21} \), we start by factoring both the numerator and the denominator. The numerator \( r^{2}-9 \) is a difference of squares, which can be factored as: \[ r^{2}-9 = (r-3)(r+3) \] For the denominator \( r^{2}-4r-21 \), we look for two numbers that multiply to -21 and add to -4. The numbers -7 and 3 work, so we factor it as: \[ r^{2}-4r-21 = (r-7)(r+3) \] Now we can rewrite the expression: \[ \frac{r^{2}-9}{r^{2}-4r-21} = \frac{(r-3)(r+3)}{(r-7)(r+3)} \] Since \( r+3 \) is common in both the numerator and denominator, we can cancel it (as long as \( r \neq -3 \)): \[ = \frac{r-3}{r-7} \] However, we must state the restrictions on the variable. The expression is undefined for values that make the denominator zero. Thus: - \( r+3 = 0 \Rightarrow r \neq -3 \) - \( r-7 = 0 \Rightarrow r \neq 7 \) So, the simplified expression is: \[ \frac{r-3}{r-7} \] with restrictions \( r \neq -3 \) and \( r \neq 7 \).