The following simultaneous nonlinear equations are to be solved:
$$y=e^x$$
$$y=x(1+x)$$
Define an objective function that can be maximized to obtain a solution to these equations.
Sketch x vs F(x) and identify the optimal solution (x*) graphically.
I don't know if I'm supposed to set the two y equations equal to each other and solve for x? How do I get an objective function from this?
Well if we want to find a solution to the equations you have given then let the objective function be
\begin{align} F(x) = -|f_1(x)-f_2(x)| \end{align}
where $f_1(x) = e^x$ and $f_2(x)= x(x+1)$. Then maximising $F$ is equivalent to minimising the distance between the two functions which will yield a solution to the system. You can quickly sketch the function in question by first sketching the function inside the absolute value sign and then reflecting about the x-axis whenever it goes below zero. Then "flip" it and you should get an idea where the solution is (the maximum of the resulting function).
