Find a $G.P.$ for which the sum of the first two terms is $-4$ and the fifth term is $4$ times the third term.

Let $a$ be the first term and $r$ be the common ratio of the $G.P.$
According to the given conditions:
$a + ar = -4$ .......$(1)$
$ar^4 = 4 \times ar^2$
Since $a \neq 0$,we have $r^2 = 4$,which implies $r = 2$ or $r = -2$.
Case $1$: If $r = 2$,then from $(1)$:
$a + 2a = -4$ $\Rightarrow 3a = -4$ $\Rightarrow a = -\frac{4}{3}$.
The $G.P.$ is $-\frac{4}{3}, -\frac{8}{3}, -\frac{16}{3}, \dots$
Case $2$: If $r = -2$,then from $(1)$:
$a - 2a = -4$ $\Rightarrow -a = -4$ $\Rightarrow a = 4$.
The $G.P.$ is $4, -8, 16, -32, \dots$