$\text{ Let }A=\left\{\text{ set of values of }x\text{ for which }{x}^2+(m-1)x-2(m+1)=0,x\in R\right\}\text{ , }$$B=\left\{\text{ set of values of }x\text{ for which }(m-1)x^2+mx+1=0,x\in R\right\}.\text{ Number of values of }m$$\text{ such that }A\cup B\text{ has exactly }3\text{ distinct elements is: }$

$x^2+\left(m-1\right)x-2\left(m+1\right)=0$

$\Rightarrow x=2orx=-\left(m+1\right)$

$\left(m-1\right)x^2+mx+1=0$

$\Rightarrow x=-1orx=-\frac{1}{\left(m-1\right)}$

Case (i) $-\left(m+1\right)=-\frac{1}{\left(m-1\right)}$$\Rightarrow m=\pm \sqrt{2}$

Case (ii) $-\left(m+1\right)=2\Rightarrow m=-3$

Case (iii) $-\left(m+1\right)=-1\Rightarrow m=0$

Case (iv)   $-\frac{1}{\left(m-1\right)}=-1\Rightarrow m=2$             

Case (v) $-\frac{1}{\left(m-1\right)}=2\Rightarrow m=\frac{1}{2}$