To find an expression equivalent to
[tex]\[ \frac{\frac{m+3}{m^2-16}}{\frac{m^2-9}{m+4}} \][/tex]
we will first simplify the given expression step-by-step.
### Step 1: Simplify the Denominators by Factoring
We start by factoring the denominators [tex]\( m^2 - 16 \)[/tex] and [tex]\( m^2 - 9 \)[/tex]:
1. [tex]\( m^2 - 16 \)[/tex] is a difference of squares and can be factored as:
[tex]\[
m^2 - 16 = (m - 4)(m + 4)
\][/tex]
2. [tex]\( m^2 - 9 \)[/tex] is also a difference of squares and can be factored as:
[tex]\[
m^2 - 9 = (m - 3)(m + 3)
\][/tex]
### Step 2: Rewrite the Expression
Using these factorizations, we rewrite the original complex fraction:
[tex]\[
\frac{\frac{m+3}{(m-4)(m+4)}}{\frac{(m-3)(m+3)}{m+4}}
\][/tex]
### Step 3: Simplify the Fraction
To divide by a fraction, we multiply by its reciprocal. Thus, the expression becomes:
[tex]\[
\frac{m+3}{(m-4)(m+4)} \times \frac{m+4}{(m-3)(m+3)}
\][/tex]
### Step 4: Cancel Common Factors
We observe that [tex]\( (m+4) \)[/tex] and [tex]\( (m+3) \)[/tex] appear in both the numerator and denominator, allowing these terms to be canceled:
[tex]\[
\frac{m+3}{(m-4)(m+4)} \times \frac{m+4}{(m-3)(m+3)} = \frac{1}{(m-4)} \times \frac{1}{(m-3)}
\][/tex]
Thus, the expression simplifies to:
[tex]\[
\frac{1}{(m-4)(m-3)}
\][/tex]
### Conclusion
The expression equivalent to the given complex fraction is
[tex]\[
\boxed{\frac{1}{(m-4)(m-3)}}
\][/tex]
Therefore, the correct answer is the second option:
[tex]\(\frac{1}{(m-4)(m-3)}\)[/tex]
