The value of m in order that $x^4-2x^3+3x^2-mx+5$ may be exactly divisible by $x-3$is

$p\left(x\right)=x^4-2x^3+3x^2-mx+5$ is exactly divisible by $x-3$

Then $p\left(3\right)=3^4-2{\left(3\right)}^3+3{\left(3\right)}^2-m\left(3\right)+5=0$

$\begin{array}{l}\Rightarrow 81-54+27-3m+5=0\\ 59-3m=0\\ m=\frac{59}{3}\end{array}$