Find Polynomial coefficients How can I find the real coefficients a,b if the polynomial 
$ P(X)=X^4-5X^3+8X^2+aX+b $ is divisible by $ Q(X)=(X-1)^2 $
So if 1 is a solution, I get a+b=-4; where should i look for other roots?

Using synthetic division:
\begin{array}{rrrrrrr}
       & | &   1 &  -5 &   8 &   a     & b \\
     1 & | &   0 &   1 &  -4 &   4 &   a+4 \\
       & + & --- & --- & --- & --- & ----- \\
       &   &   1 &  -4 &   4 & a+4 & a+b+4 \\
\end{array}
So $a+b+4 = 0$.
Dividing again by $(x-1)$ :
\begin{array}{rcrrrr}
       & | &   1 &  -4 &   4 & a+4 \\
     1 & | &   0 &   1 &  -3 &   1 \\
       & + & --- & --- & --- & --- \\
       &   &   1 &  -3 &   1 & a+5 \\
\end{array}
So $a+5= 0$, hence $a = -5$ and $b=1$.
