Find the terms of a geometric progression given two partial sums.

Hint:

\begin{align} a_1+a_2+a_3+a_4+a_5&=93\\ a_1+(a_2+a_4)(1+q)&=93\\ a_1+30(1+q)&=93\\ a_1&=63-30q \end{align}

Then:

\begin{align} a_2+a_4&=30\\ a_1q(1+q^2)&=30\\ (63-30q)q(1+q^2)&=30\\ (21-10q)q(1+q^2)&=10\\ \iff 10q^4-21q^3+10q^2-21q+10&=0 \end{align}

Now try with the Rational Root Theorem.