If $g\left(x\right)$  is a polynomial function satisfying  $g\left(x\right)g\left(y\right)=g\left(x\right)+g\left(y\right)+g\left(xy\right)-2$, for all  $x,y\in R$ and  $g\left(2\right)=5,$ then  $g\left(5\right)$ is

$g\left(x\right)g\left(y\right)=g\left(x\right)+g\left(y\right)+g\left(xy\right)-2$

Put x=1 and y=2

$g\left(1\right)g\left(2\right)=g\left(1\right)+g\left(2\right)+g\left(2\right)-2$

$5g\left(1\right))=g(1)+5+5-2$

$g\left(1\right)=2$

Put $y=\frac{1}{x}$ then

$g\left(x\right)g\left(\frac{1}{x}\right)=g\left(x\right)+g\left(\frac{1}{x}\right)+g\left(1\right)-2$

$g\left(x\right)g\left(\frac{1}{x}\right)=g\left(x\right)+g\left(\frac{1}{x}\right)$

which satisfy the function is

$g\left(x\right)=x^2+1$

Therefore, $g\left(2\right)=2^2+1=5$ and $g\left(5\right)=5^2+1=25+1=26$