A patient is diagnosed to have $P_0$ cancer cells at time $t=0$ and these cells grow subsequently according to $\frac{dP}{dt}=-\alpha P(M-P)$ where $P(t)$ is the number of cells at time $t$ while $\alpha$ and $M$ are two physiologically significant positive constants. The patient will die when the number of cancer cells approaches infinity. Compute the condition between the values $P_0$ and $M$ when the person will die because of this cancer. Find the time the patient has left to live starting from the day of diagnosis.
Solving the DE, I get $t = \frac{1}{\alpha (M-1)P}$ or $P(t) = \frac{1}{\alpha t(M-1)}$
So, shouldn't the patient die when $M$ approaches infinity?
How could I compute the patient's remaining time?
I'm not sure how you got your solution $P(t)$. Divide both sides by $P(M-P)$ and then apply partial fraction decomposition to evaluate the integral. The correct answer is
$$P(t)=\frac{Me^{cM}}{e^{cM}-e^{aMt}},$$
where
$$\frac{e^{cM}}{e^{cM}-1}=P_0,$$
which you can solve for $e^{cM}$ (or $c$).
Now notice $e^{aMt}=e^{cM}$ causes blowup, so just finish this analysis up by checking for which $c,M$ this is possible. Notice that this happens in finite time. It doens't look like it's possible for it to occur in infinite time.