\begin{align*} \text{Numerator = } & E[X^4] - 4 \mu E[X^3] + 6 \mu^2 E[X^2] - 3 \mu^4\\ & = \int_{-1}^1 x^4 \cdot \frac{3}{4} \cdot (1-x^2) dx\\ & = \frac{3}{4} \cdot (x^5 /5-x^7 /7) \big |_{-1}^1\\ & = \frac{3}{4} \cdot 2(1/5-1/7)\\ & = \frac{3}{35} \end{align*}