$$\begin{gathered} \text{ If }f\left(x\right)=\left(x^2+3x+2\right)\left(x^2-7x+a\right)\text{ and }g\left(x\right)=\left(x^2-x-12\right)\left(x^2+5x+b\right),\text{ then the values of }a\text{ and }b\text{ , if }(x+1)(x-4) \\ \text{ is HCF of }f\left(x\right)\text{ and }g\left(x\right),\text{ are } \end{gathered}$$

$$\begin{gathered} f\left(x\right)=\left(x^2+3x+2\right)\left(x^2-7x+a\right) \\ =(x+1)(x+2)\left(x^2-7x+a\right) \end{gathered}$$

f(x) is divisible by (x+1)(x-4)

$x^2-7x+a\text{ is divisible by }x-4$  $\Rightarrow a=12$

similarly $g\left(x\right)=\left(x^2-x-12\right)\left(x^2+5x+b\right)$

                      =$(x-4)(x+3)(x^2+5x+b)$

$x^2+5x+b\text{ is divisible by }x+1$

$\Rightarrow b=4$