Simplify. \[ \frac{2 w^{2}+19 w+35}{4 w^{2}-25} \]

To simplify the expression \(\frac{2 w^{2}+19 w+35}{4 w^{2}-25}\), we will first factor both the numerator and the denominator. 1. Factor the numerator \(2w^2 + 19w + 35\): - We need two numbers that multiply to \(2 \times 35 = 70\) and add up to \(19\). - These numbers are \(14\) and \(5\). - We can rewrite the middle term using these numbers: \[ 2w^2 + 14w + 5w + 35 \] - Now factor by grouping: \[ 2w(w + 7) + 5(w + 7) = (2w + 5)(w + 7) \] 2. Factor the denominator \(4w^2 - 25\): - This is a difference of squares, which factors as follows: \[ 4w^2 - 25 = (2w - 5)(2w + 5) \] Now we can rewrite the original expression with the factored forms: \[ \frac{(2w + 5)(w + 7)}{(2w - 5)(2w + 5)} \] Now, observe that the factor \(2w + 5\) in the numerator and denominator cancels out (as long as \(2w + 5 \neq 0\)). Thus, we have: \[ \frac{w + 7}{2w - 5} \] Final answer: \[ \boxed{\frac{w + 7}{2w - 5}} \]