Simplify. \( \frac{w^{2}-9 w+14}{16-4 w^{2}} \)

To simplify the expression \( \frac{w^{2}-9 w+14}{16-4 w^{2}} \), let's start by factoring both the numerator and the denominator. The numerator \( w^{2}-9w+14 \) factors as \( (w-7)(w-2) \) because \( -7 \) and \( -2 \) are the roots of the quadratic. Now, let’s factor the denominator \( 16-4w^{2} \). We can rewrite it as \( 4(4-w^{2}) \), and then apply the difference of squares: \( 4(2-w)(2+w) \). Putting it all together, we have: \[ \frac{(w-7)(w-2)}{4(2-w)(2+w)} \] Note that \( (2-w) \) is the negative of \( (w-2) \): \( (2-w) = -(w-2) \). Incorporating this fact into our expression, we get: \[ \frac{(w-7)(w-2)}{4(-1)(w-2)(2+w)} = \frac{-(w-7)}{4(2+w)} \] Final simplified expression: \[ \frac{7-w}{4(2+w)} \]