Given, $f\left(g\left(x\right)=8x^2-2x\right.$and  $g\left(f\left(x\right)=4x^2+6x+1\right.$

Let $f\left(x\right)=c{x}^2+dx+e$ and $g\left(x\right)=ax+b$

So, $f\left(g\left(x\right)\right)=c{\left(ax+b\right)}^2+d\left(ax+b\right)+e=8x^2-2x$

and $g\left(f\left(x\right)\right)=a\left(c{x}^2+dx+e\right)+b=4x^2+6x+1$

On comparing both side we get, $a=2, b=-1, c=2, d=3, e=1$

So, $g\left(x\right)=2x-1\Rightarrow g\left(2\right)=3$

$\& f\left(x\right)=2x^2+3x+1$

$f\left(2\right)=8+6+1=15$.

So, $f\left(2\right)+g\left(2\right)=18$