\( 80 q^{3}+32 q^{2}-5 q-2=0 \) he solution set is \( \{\square\} \).

To solve the cubic equation \( 80 q^{3}+32 q^{2}-5 q-2=0 \), we can utilize numerical methods or graphing techniques. However, factoring or applying the Rational Root Theorem first could also help find rational roots that simplify our computations. Once potential rational roots are spotted, we can verify each one by substitution, and then use polynomial long division to break down the cubic polynomial into simpler factors. Like many cubic equations, this one can exhibit interesting behaviors, such as having one real root and a pair of complex conjugate roots! Exploring these properties can take you into the realms of complex numbers and even help you understand the fundamental theorem of algebra better.