Find the minimum value of 2x^2 + 2xy + y^2 - 2x + 2y + 4 - x^2 + 4x over all real numbers $x$ and $y.$
+2666 Rearrange as: \( (y^2 + 2y) +(x^2 +2x+ 2xy) + 4 \)
Consider this as 2 separate equations (\(x^2 +2x+ 2xy\) and \(y^2 + 2y\)) and a constant. Our goal is to minimize the equations.
Start with \(y^2 + 2y\). The minimum value of this is -1, and it occurs when \(y = {-b \over 2a} = {-2 \over 2} = -1\)
Now, subbing in \(y = -1\) into the second equation gives us \(x^2 + 2x - 2x = x^2\).
The minimum value of this, quite obviously, is 0, and it occurs when \(x = 0\).
This means the minimum value is: \(0 - 1 + 4 = \color{brown}\boxed{3}\)
