d. \( \frac{8}{m+5}=m+5 \)

To solve the equation \( \frac{8}{m+5}=m+5 \), we can start by cross-multiplying to eliminate the fraction. This gives us \( 8 = (m+5)(m+5) \) or \( 8 = (m+5)^2 \). Expanding the right side yields: \[ 8 = m^2 + 10m + 25 \] Next, we rearrange the equation to set it to zero: \[ m^2 + 10m + 25 - 8 = 0 \implies m^2 + 10m + 17 = 0 \] Now, we can use the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 10, c = 17 \): \[ b^2 - 4ac = 10^2 - 4 \cdot 1 \cdot 17 = 100 - 68 = 32 \] Now plug the values into the quadratic formula: \[ m = \frac{-10 \pm \sqrt{32}}{2 \cdot 1} = \frac{-10 \pm 4\sqrt{2}}{2} = -5 \pm 2\sqrt{2} \] Thus, the solutions are: \[ m = -5 + 2\sqrt{2} \quad \text{and} \quad m = -5 - 2\sqrt{2} \] These two values for \( m \) are your final answers!