Find for which real parameter a matrix is diagonalisable

Note that:

$$-\lambda^3+(3+h)\lambda^2 - (2+3h)\lambda + 2h=-\lambda^3+3\lambda^2 -2\lambda +h\lambda^2-3h\lambda + 2h=-\lambda(\lambda^2-3\lambda +2)+h(\lambda^2-3\lambda +2)=(h-\lambda)(\lambda^2-3\lambda +2)=(h-\lambda)(\lambda-1)(\lambda-2)$$

The characteristic polynomial of $A$ is$$-x^3+(h+3)x^2-(3h+2)x+2h,$$whose discriminant is$$h^4-6h^3+13h^2-12h+4=(h^2-3h+2)^2=(h-1)^2(h-2)^2.$$Therefore, if $h\notin\{1,2\}$, then $A$ has $3$ distinct eigenvalues and then it must be diagonalizable. All you have to do is to see what happens whan $h\in\{1,2\}$.