Final answer:
The simplification of the given algebraic expression involves factoring and then dividing fractions, which results in the expression \((n-6)/(6(n-5))\).
Step-by-step explanation:
The question at hand is a mathematical expression simplification problem that involves quadratic equations and factoring. To simplify the given expression, we will need to first factor both the numerator and denominator of each fraction and then perform division of the two fractions. None of the reference information provided explicitly matches the expressions given in the student's question, but we can apply general simplification methods.
Let's simplify the expression \(\frac{n^2 - 4n - 12}{6n - 30} \div \frac{n^2 + 10n + 16}{n + 8}\). We start by factoring the quadratics and the linear terms where possible:
- \(n^2 - 4n - 12\) factors into \((n - 6)(n + 2)\)
- \(6n - 30\) is \(6(n - 5)\)
- \(n^2 + 10n + 16\) factors into \((n + 2)(n + 8)\)
- \(n + 8\) remains as is since it's already a linear factor
Now, plug these factors back into the original expression and simplify:
\(\frac{(n - 6)(n + 2)}{6(n - 5)} \div \frac{(n + 2)(n + 8)}{n + 8}\)
Dividing by a fraction is the same as multiplying by its reciprocal, so we multiply by the reciprocal of the second fraction:
\(\frac{(n - 6)(n + 2)}{6(n - 5)} \times \frac{n + 8}{(n + 2)(n + 8)}\)
Now we can cancel out the common factors \((n + 2)\) and \((n + 8)\):
\(\frac{n - 6}{6(n - 5)}\)
This is the simplified expression.