a) do you know what it means for a function to be analytic?
b) do you know what the Cauchy–Riemann equations are?
if not read up on those two and give it a shot.
you have u so compute it's partials
du/dx = 2x
du/dy = 2y
so dv/dy = 2x and thus v = 2xy + f(y)
dv/dx = -du/dy = -2y so v = -2xy + g(x)
2xy + f(y) = -2xy + g(x)
4xy = g(x) - f(y), and you can see that this has no solution