$\mathrm{If} {\mathrm{x}}^2-6\mathrm{x}+\mathrm{p} \mathrm{and} {\mathrm{x}}^2-8\mathrm{x}-\mathrm{q} \mathrm{have} \mathrm{a} \mathrm{common} \mathrm{factor} \mathrm{then} \mathrm{find} \mathrm{the} \mathrm{factor}$

Let  x - k  is the common factor
By the factor theorem

$$\begin{gathered} f\left(k\right)=k^2-6k+p=0 \\ g\left(k\right)=k^2-8k-q=0 \\ \Rightarrow k^2-6k+p=k^2-8k-q \\ 2k=-(p+q) \\ k=-\left(\frac{p+q}{2}\right) \\ \therefore The common factor is x+\left(\frac{p+q}{2}\right). \end{gathered}$$