\( \mid \leftarrow \quad \) Perform the indicated operation. Write all answers in lowest terms. \[ \frac{y^{2 n}-y^{n}-12}{5 y^{n}-20} \div \frac{y^{2 n}-9}{3+y^{n}} \]

To solve the expression \[ \frac{y^{2 n}-y^{n}-12}{5 y^{n}-20} \div \frac{y^{2 n}-9}{3+y^{n}}, \] we will follow these steps: 1. **Rewrite the division as multiplication by the reciprocal**: \[ \frac{y^{2 n}-y^{n}-12}{5 y^{n}-20} \times \frac{3+y^{n}}{y^{2 n}-9}. \] 2. **Factor each part of the expression**. ### Step 1: Factor the Numerators and Denominators **Numerator of the first fraction**: \(y^{2n} - y^n - 12\) Let \(x = y^n\). Then we can rewrite it as: \[ x^2 - x - 12. \] Factoring gives: \[ (x - 4)(x + 3) = (y^n - 4)(y^n + 3). \] **Denominator of the first fraction**: \(5y^n - 20\) Factoring out the common term: \[ 5(y^n - 4). \] **Numerator of the second fraction**: \(3 + y^n\) This is already in its simplest form. **Denominator of the second fraction**: \(y^{2n} - 9\) This is a difference of squares: \[ (y^n - 3)(y^n + 3). \] ### Step 2: Substitute the factored forms back into the expression Now substituting the factored forms, we have: \[ \frac{(y^n - 4)(y^n + 3)}{5(y^n - 4)} \times \frac{3 + y^n}{(y^n - 3)(y^n + 3)}. \] ### Step 3: Cancel common factors Notice that \(y^n - 4\) appears in both the numerator and denominator, so we can cancel it: \[ \frac{(y^n + 3)}{5} \times \frac{3 + y^n}{(y^n - 3)(y^n + 3)}. \] Now, \(3 + y^n\) is the same as \(y^n + 3\), so we can cancel \(y^n + 3\): \[ \frac{1}{5} \times \frac{1}{y^n - 3}. \] ### Final Result Thus, the expression simplifies to: \[ \frac{1}{5(y^n - 3)}. \] This is the final answer in lowest terms.