Simplify. \( -\frac{5-3 e}{4-2 e} \pm \sqrt{\left(\frac{5-3 e}{4-2 e}\right)^{2}-\frac{3}{e-2}} \)

To simplify \( -\frac{5-3 e}{4-2 e} \pm \sqrt{\left(\frac{5-3 e}{4-2 e}\right)^{2}-\frac{3}{e-2}} \), let’s break it down step by step. First, let's simplify the expression inside the square root: \[ \left(\frac{5-3e}{4-2e}\right)^{2} - \frac{3}{e-2} \] Finding a common denominator for the two fractions will help simplify: The common denominator between \( (4-2e)^2 \) and \( e-2 \) is \( (4-2e)^2(e-2) \). So we can write: \[ \left(\frac{(5-3e)^2}{(4-2e)^2}\right) - \frac{3(4-2e)^2}{(e-2)(4-2e)^2} \] Next, we proceed to add them together: \[ \frac{(5-3e)^2 - 3(4-2e)^2}{(4-2e)^2} \] Now, calculate \( (5-3e)^2 \) and \( 3(4-2e)^2 \). Expanding \( (5-3e)^2 = 25 - 30e + 9e^2 \) and \( 3(4-2e)^2 = 3(16 - 16e + 4e^2) = 48 - 48e + 12e^2 \). Putting it all together, we have: \[ \left(25 - 30e + 9e^2\right) - \left(48 - 48e + 12e^2\right) = -23 + 18e - 3e^2 \] Now, we need to combine this and proceed with evaluating: Finally, we substitute back into the original simplified expression. Thus, the expression simplifies to: \[ -\frac{5-3e}{4-2e} \pm \sqrt{\frac{-3e^2 + 18e - 23}{(4-2e)^2}} \] This representation helps in understanding the simplification without losing the essence of the square root in the expression.