composition of function backwards

Question

If $f\circ g=h, h(x)=2x^2+2x+2, f(x)=2x+4$ what is $g(x)?$

I do not even know where to begin. I think you have to use inverse of one of the functions but im not sure. please help

Answer 1

Notice, we have $h(x)=2x^2+2x+2$

$$f(x)=2x+4$$ $$\implies (fog)(x)=f(g(x))=2g(x)+4\tag 1$$

Also given $$(fog)(x)=h(x)$$$$\implies (fog)(x)=2x^2+2x+2\tag 2$$ Now, equating (1) & (2), we get $$2g(x)+4=2x^2+2x+2$$ $$2g(x)=2x^2+2x-2$$ $$\color{red}{g(x)=x^2+x-1}$$

Answer 2

$$2x^2+2x+2=h(x)=(f\circ g)(x)=f(g(x))=2g(x)+4$$

Can you solve for $g(x)$ now?