$x^4-2x^2=x^2(x^2-2)\leq0$ when $|x|\leq\sqrt{2}$ and $\geq0$ outside.

To minimise the integral (make it most negative), integrate over the region where the integrand is most negative — i.e., $[-\sqrt{2},\sqrt{2}]$ where the function is $\leq0$.

Any sub-interval of $[-\sqrt{2},\sqrt{2}]$ gives a more negative (or equal) value, and the widest such interval $[-\sqrt{2},\sqrt{2}]$ gives the minimum.

$(a,b)=(-\sqrt{2},\sqrt{2})$