Simplify: \( \frac{5 x^{2}-13 x+6}{5 x^{2}-9 x-2} \div \frac{5 x^{2}-23 x+12}{x^{2}-16} \)

To simplify the expression \[ \frac{5 x^{2}-13 x+6}{5 x^{2}-9 x-2} \div \frac{5 x^{2}-23 x+12}{x^{2}-16}, \] we first rewrite the division as multiplication by the reciprocal: \[ \frac{5 x^{2}-13 x+6}{5 x^{2}-9 x-2} \times \frac{x^{2}-16}{5 x^{2}-23 x+12}. \] Now we will factor each polynomial: 1. **Factoring the numerator \(5x^2 - 13x + 6\)**: We find two numbers that multiply to \(5 \times 6 = 30\) and add to \(-13\). The pair \(-3\) and \(-10\) works: \[ 5x^2 - 10x - 3x + 6 = (5x^2 - 10x) + (-3x + 6) = 5x(x - 2) - 3(x - 2) = (5x - 3)(x - 2). \] 2. **Factoring the denominator \(5x^2 - 9x - 2\)**: We need numbers that multiply to \(5 \times -2 = -10\) and add to \(-9\). The pair \(-10\) and \(1\) works: \[ 5x^2 - 10x + x - 2 = 5x(x - 2) + 1(x - 2) = (5x + 1)(x - 2). \] 3. **Factoring the numerator \(5x^2 - 23x + 12\)**: Here, we need numbers that multiply to \(5 \times 12 = 60\) and add to \(-23\). The pair \(-3\) and \(-20\) works: \[ 5x^2 - 20x - 3x + 12 = (5x^2 - 20x) + (-3x + 12) = 5x(x - 4) - 3(x - 4) = (5x - 3)(x - 4). \] 4. **Factoring the denominator \(x^2 - 16\)**: This is a difference of squares: \[ x^2 - 16 = (x - 4)(x + 4). \] Now substituting all the factored forms back into the expression, we have: \[ \frac{(5x - 3)(x - 2)}{(5x + 1)(x - 2)} \times \frac{(x - 4)(x + 4)}{(5x - 3)(x - 4)}. \] Next, we can cancel common factors: - The factor \(5x - 3\) cancels. - The factor \(x - 2\) cancels. - The factor \(x - 4\) cancels. Thus the simplified expression becomes: \[ \frac{(x + 4)}{(5x + 1)}. \] Therefore, the final answer is: \[ \boxed{\frac{x + 4}{5x + 1}}. \]