The value of k so that ${\mathrm{x}}^4 – 3{\mathrm{x}}^3+ 5{\mathrm{x}}^2 – 33\mathrm{x} + \mathrm{k}$ is divisible by ${\mathrm{x}}^2 – 5\mathrm{x} + 6$ is

$\begin{array}{l}{x}^4-3x^3+5x^2-33x+kisdivisibleby{x}^2-5x+6\\ \therefore x^2-5x+6=(x-2)(x-3)\\ \therefore x-2isfactorof{x}^4-3x^3+5x^2-33x+k\\ \Rightarrow {\left(2\right)}^4-3{\left(2\right)}^3+5{\left(2\right)}^2-33\left(2\right)+k=0\\ \Rightarrow 16-24+20-66+k=0\\ \Rightarrow k-54=0\\ \Rightarrow k=54\end{array}$