Find the minimum value of the expression $x^2+y^2+2x-4y+8+10x-12y$ for real $x$ and $y$.
Simplify as
x^2 + 12x + y^2 - 16y + 8
Take the derivative with respect to x and set to 0
2x + 12 = 0
x = -6
Take the derivative with respect to y and set to 0
2y - 16 = 0
y = 8
Minimum is
(-6)^2 + 12 (-6) + (8)^2 - 16 (8) + 8 = -92
+1678 
