if we know $\displaystyle \psi(x,0)=cx(a-x) $
how to find c
I guess the well has borders at 0 and a. Hence we know (with 100% probability) that particle is inside the well. Thus probability that particle is inside the well is 1:
$\displaystyle \int_{0}^{a} \bigl|\psi(x)\bigr|^2dx = 1$
We get
$\displaystyle c^2 \int_{0}^{a} x^2(a - x)^2 dx = 1$