\(\frac{q^{2}+11q+24}{q^{2}-5q-24}\)
To simplify this expression, we factor the numerator and the denominator:
\(q^{2}+11q+24 = (q+3)(q+8)\)
\(q^{2}-5q-24 = (q-8)(q+3)\)
Now we can rewrite the expression with these factors:
\(\frac{(q+3)(q+8)}{(q-8)(q+3)}\)
Since we have a common factor of \(q+3\) in the numerator and the denominator, we can cancel them out:
\(\frac{q+8}{q-8}\)
So, the simplified form of the expression \(\frac{q^{2}+11q+24}{q^{2}-5q-24}\) is \(\frac{q+8}{q-8}\).