The factors of $f\left(x\right)=x^2-5x+6$ are:

Given:

The polynomial $f\left(x\right)=x^2-5x+6$

To factorise it, we will split the middle term as below

$\begin{array}{rcl}f\left(x\right)& =& x^2-5x+6\\ & =& x^2-2x-3x+6\\ & =& x(x-2)-3\left(x-2\right)\\ & =& \left(x-3\right)\left(x-2\right)\end{array}$

Hence, the factors of the given polynomial are $\left(x-3\right) \mathrm{and} \left(x-2\right)$.

Therefore, option C is correct.