Simplify. Express your answer as a single fraction in simplest form. \( \frac{10}{8 w^{2}-16 w-6}+6 w+6 \)

To simplify the expression \( \frac{10}{8 w^{2}-16 w-6}+6 w+6 \), we start by factoring the quadratic in the denominator. First, the quadratic \( 8w^2 - 16w - 6 \) can be factored out. To do this, we can take out a common factor of 2: \[ 8w^2 - 16w - 6 = 2(4w^2 - 8w - 3) \] Next, we will factor \( 4w^2 - 8w - 3 \) using the quadratic formula, \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 4, b = -8, c = -3 \): \[ b^2 - 4ac = (-8)^2 - 4 \cdot 4 \cdot (-3) = 64 + 48 = 112 \] Thus, the solutions for \( w \) are: \[ w = \frac{8 \pm \sqrt{112}}{8} = \frac{8 \pm 4\sqrt{7}}{8} = 1 \pm \frac{\sqrt{7}}{2} \] Therefore, we can rewrite \( 4w^2 - 8w - 3 \) as \((w - (1 + \frac{\sqrt{7}}{2}))(w - (1 - \frac{\sqrt{7}}{2}))\). Now our expression becomes: \[ \frac{10}{2(4w^2 - 8w - 3)} + 6w + 6 \] This simplifies to: \[ \frac{5}{4w^2 - 8w - 3} + 6w + 6 \] To combine these, we need a common denominator, which is \( 4w^2 - 8w - 3 \). We rewrite \( 6w + 6 \) in terms of this denominator: \[ 6w + 6 = \frac{6w + 6(4w^2 - 8w - 3)}{4w^2 - 8w - 3} = \frac{6w + 24w^2 - 48w - 18}{4w^2 - 8w - 3} \] Simplifying the numerator gives: \[ 24w^2 - 42w - 18 \] Now, we can combine the fractions: \[ \frac{5 + 24w^2 - 42w - 18}{4w^2 - 8w - 3} = \frac{24w^2 - 42w - 13}{4w^2 - 8w - 3} \] The final answer, expressed as a single fraction in simplest form, is: \[ \frac{24w^2 - 42w - 13}{4w^2 - 8w - 3} \]