If we rewrite the above equation as $x^3 - ax^2 + 23x - b = (x-M)^2(x-N) = 0$ such that $M+N=12$ (i), we obtain: $(x-M)^2(x-N) = x^3 - (2M+N)x^2 + (M^2+2MN)x - M^2N = x^3 - ax^2 + 23x - b \Rightarrow M^2+2MN=23$ (ii); and solving the system of (i) and (ii) yields: $M^2 + 2M(12-M)=23 \Rightarrow -M^2 + 24M - 23 = -(M-1)(M-23) = 0 \Rightarrow M = 1, 23 \Rightarrow (M, N) = (1, 11); (23, -11)$. We finally compute: $a = 2M+N \Rightarrow$$\fbox{a = 13, 35}$. $b = M^2N \Rightarrow$$\fbox{b = 11, -5819}$.