I wonder what "your level" is, as that would aid explanation. Anyway...
1) Form the new function g(x,y) which is the constraint with zero on the rhs
2) Form the new function h(x,y,a) = f(x,y) - a g(x,y)
3) Differentiate h(x,y,a) partially wrt x, then wrt y, then wrt a, set all three derivatives = 0 and solve; the stationary points are the max/min of your function subject to your constraint (note the constraint is enforced via the derivative wrt a = 0)
4) Note that the method generalises to functions of as many variables as you want
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In your example
1)
g(x,y) = y^2 - x
2)
h(x,y,a) = -x + y +xy - y^2 s - a(y^2 - x)
3) left as an exercise
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You can check your answer in this case by noticing that along the curve specified by your constraint f(x,y) = -x + y +xy - y^2 = f(x) = -x + x^{1/2} + x^{3/2} - x and differentiating normally wrt x. The point of Lagrange multpliers is that by using the method you can avoid the horrible algebra that would sometimes be involved in back-substituting your constraint into your function.