[ Methods for solving problems with parameters ] [

Specify all points $(x, y)$ in the plane through which at least one curve of the family $y=p^{2}+(2 p-1) x+$ $2 x^{2}$ passes.

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The point $(x, y)$ lies on one of the given curves if and only if the equation $p^{2}+2 p x+\left(2 x^{2}-x-y\right)=$ 0 has a solution (with respect to $p$), which means that $D / 4=x^{2}-\left(2 x^{2}-x-y\right)=y+x-x^{2} \geq 0$.

## Answer

All points above the parabola $y=x^{2}-x$, including the parabola itself.