If $\mathrm{f}\left(\mathrm{x}\right)=\left({\mathrm{x}}^2+3\mathrm{x}+2\right)\left({\mathrm{x}}^2-7\mathrm{x}+\mathrm{a}\right)$ and $\mathrm{g}\left(\mathrm{x}\right)=\left({\mathrm{x}}^2-\mathrm{x}-12\right)\left({\mathrm{x}}^2+5\mathrm{x}+\mathrm{b}\right)$ then the values of a and b, If (x+1)(x–4) is H.C.F of f(x) and g(x)

$$\begin{gathered} \begin{array}{r}f\left(x\right)=\left(x^2+3x+2\right)\left(x^2-7x+a\right)\\ =(x+1)(x+2)\left(x^2-7x+a\right)\end{array} \\ f\left(x\right)\text{ is divisible by }(x+7)(x-4) \\ {x}^2-7x+a\text{ is divisible by }x-4\Rightarrow a=12 \\ \begin{array}{r}\text{ similarly }g\left(x\right)=\left(x^2-x-12\right)\left(x^2+5x+b\right)\\ =(x-4)(x+3)\left(x^2+5x+b\right)\end{array} \end{gathered}$$

$$\begin{gathered} x^2+5x+b\text{ is divisible by }x+1 \\ b=4 \end{gathered}$$