have you successfully drawn the graph? that's pretty much the hard part... and idk if there is anyway to solve for points of intersection besides inspecting the equation e^(x+1)=x^2 and the graph ... but yeah, (-1,1) is the only point of intersection
From there you can find the area by substracting the area under y=x^2 between -1 and 0 from the area under y=e^(x+1) between -1 and 0. answer become (e-4/3) units^2