So "Do my homework for me"? No, that is not going to happen! (You don't even say "please"!)
Have you graphed these so you can see what region you are working with and what the limits of integration should be? The first problem has one boundary $x^2+ y= 4$. That is the same as $y= 4- x^2$, a parabola. its "vertex" is at (0, 4) and it crosses the x-axis at (2, 0) and (-2, 0). The next is y+ x- 2= 0 which is the same as y= 2- x. That's a straight line. When x= 0, y= 2- 0= 2 and when y= 0, 2- x= 0 so x= 2. Draw the straight line through (0, 2) and (2, 0). Do you see that the line crosses the parabola where $y= 4- x^2= 2- x$. $x^2- x- 2= (x- 2)(x+ 1)= 0$ so the line crosses the parabola at (-1, 3) and (2, 0). The other two boundaries are the vertical lines x= -2 and x= 3.
Frankly that makes no sense at all! The last two vertical lines are completely outside the region bounded by the first two so there is NO region bounded by these four. If I were forced to give a numeric answer I would give the area of the region bounded by $y= 4- x^2$ and $y= 2- x$. That region goes from x= -1 on the left to x= 2 on the right and, for each x. from y= 2- x below to $y= 4- x^2$.
Imagine dividing that region into thin vertical strips of width "dx". The length of each strip is $4- x^2- (2- x)= 2+ x- x^2$ so each strip has area $(2+ x- x^2)dx$. integrate that from x= -1 to x= 2.