We know that ;

$\frac{1}{1-x}=1+ x+x^{2}+x^{3}+x^{4}+x^{5}+...................$

Putting $x^{2}$ in place of x we get ,

$\frac{1}{1-x^{2}}=1+ x^{2}+x^{4}+x^{6}+x^{8}+x^{10}+...................$

Given ,

$G(x)=x^{2}+3x+7+\frac{1}{1-x^{2}}$

$G(x)=x^{2}+3x+7+1+x^2+x^4+x^{6}+x^8$

$G(x)=8+3x+2x^2+0.x^3+x^4+0.x^5+x^{6}+0.x^7+x^8+x^{10}+x^{12}+........$

So, $a_{0}=8$ , $a_{7}=0$ ; $a_{0}$, $a_{7}$ are coefficient of $x^0$ and $x^7$ .

$\frac{1}{1-x}=1+ x+x^{2}+x^{3}+x^{4}+x^{5}+...................$

Putting $x^{2}$ in place of x we get ,

$\frac{1}{1-x^{2}}=1+ x^{2}+x^{4}+x^{6}+x^{8}+x^{10}+...................$

Given ,

$G(x)=x^{2}+3x+7+\frac{1}{1-x^{2}}$

$G(x)=x^{2}+3x+7+1+x^2+x^4+x^{6}+x^8$

$G(x)=8+3x+2x^2+0.x^3+x^4+0.x^5+x^{6}+0.x^7+x^8+x^{10}+x^{12}+........$

So, $a_{0}=8$ , $a_{7}=0$ ; $a_{0}$, $a_{7}$ are coefficient of $x^0$ and $x^7$ .