Answer: \(-2\leqslant x\leqslant\frac54\).

The critical values occur when one of the factors is zero:

\((x+2)(4x-5)=0.\)

So

\(x=-2\quad\text{or}\quad x=\frac54.\)

The product \((x+2)(4x-5)\) is a quadratic with positive leading coefficient, so it is positive outside the roots and non-positive between the roots.

Because the inequality is

\((x+2)(4x-5)\leqslant0,\)

we take the interval between the two critical values, including both endpoints.

Therefore

\(-2\leqslant x\leqslant\frac54.\)