If $(\mathrm{x}-2)$ is a common factor of the expression ${\mathrm{x}}^2+\mathrm{ax}+\mathrm{b}$ and ${\mathrm{x}}^2+\mathrm{cx}+\mathrm{d},$ then $\frac{\mathrm{b}-\mathrm{d}}{\mathrm{c}-\mathrm{a}}=$

$\begin{array}{l}Ifx-2isfactorof{x}^2+ax+band{x}^2+cx+d\\ thenc=2isrootof{x}^2+ax+band{x}^2+cx+d\\ \Rightarrow 4+2a+b=0--\left(1\right)and4+2c+d=0--\left(2\right)\\ \left(1\right)-\left(2\right)\Rightarrow 2(a-c)+(b-d)=0\\ \Rightarrow b-d=2(c-a)\\ \Rightarrow \frac{b-d}{c-a}=2\end{array}$