I'm to solve the equation
$\ln(9t+45) - \ln(5-t) = \ln(t+3)^2$
After some work I arrive at this:
$t(t^3-4t^2+25t-35) = 0$
which clearly shows that $0$ is a root for $t$. This is also clear when testing:
$\ln(0+45) - \ln(5-0) = \ln(0+3)^2 \iff \ln\frac{45}{5} = \ln9$
But how do I know this is the only solution? I can't see any way of factorising $t(t^3-4t^2+25t-35)$ any further, and I can't see any obvious roots. I can't just assume $t=0$ is the only solution, can I?