If $f(x)=x^{2}+g^{\prime}(1) x+g^{\prime \prime}(2)$ and $g(x)=f(1) x^{2}+x f^{\prime}(x)+f^{\prime \prime}(x)$, then the value of $f(4)-g(4)$ is equal to ____________.

Let $g^{\prime}(1)=a$ and $g^{\prime \prime}(2)=b$ <br/><br/>$\Rightarrow f(x)=x^{2}+a x+b$ <br/><br/>Now, $f(1)=1+a+b ; f^{\prime}(x)=2 x+a ; f^{\prime \prime}(x)=2$ <br/><br/>$g(x)=(1+a+b) x^{2}+x(2 x+a)+2$ <br/><br/>$\Rightarrow g(x)=(a+b+3) x^{2}+a x+2$ <br/><br/>$\Rightarrow g^{\prime}(x)=2 x(a+b+3)+a \Rightarrow g^{\prime}(1)=2(a+b+3)$$+a=a$ <br/><br/>$\Rightarrow a+b+3=0$ .......(1) <br/><br/>$g^{\prime \prime}(x)=2(a+b+3)=b$ <br/><br/>$\Rightarrow 2 a+b+6=0$ ........(2) <br/><br/>Solving (i) and (ii), we get <br/><br/>$a=-3$ and $b=0$ <br/><br/>$f(x)=x^{2}-3 x$ and $g(x)=-3 x+2$ <br/><br/>$f(4)=4$ and $g(4)=-12+2=-10$ <br/><br/>$\Rightarrow f(4)-g(4)=16-2=14$