Answer
$t = 0, -1, -5$
Work Step by Step
$t^{3} +6t^{2} = - 5t$
$t^{3} +6t^{2} + 5t = 0$
$t[t^{2} + 6t + 5] = 0$
$t[t^{2} + 5t + t + 5] = 0$
$t[t(t+5)+1(t+5)] = 0$
$t(t+1)(t+5) = 0 $
$t = 0, -1, -5$
Check:
When $t = 0$
$t^{3} +6t^{2} \overset{?}{=} - 5t$
$0^{3} +6(0)^{2} \overset{?}{=} - 5(0)$
$0 +0 \overset{?}{=} (0)$
$0 = 0$
When $t = -1$
$t^{3} +6t^{2} \overset{?}{=} - 5t$
$(-1)^{3} +6(-1)^{2} \overset{?}{=} - 5(-1)$
$-1 +6(1) \overset{?}{=} 5$
$-1 +6 \overset{?}{=} 5$
$5 = 5$
When $t = -5$
$t^{3} +6t^{2} \overset{?}{=} - 5t$
$(-5)^{3} +6(-5)^{2} \overset{?}{=} - 5(-5)$
$-125 +6(25) \overset{?}{=} 25$
$-125 +150 \overset{?}{=} 25$
$25 = 25$
