Find the least value of the expression $x^2+2xy+2y^2+4y+7$ Find the least value of the expression
$x^2+2xy+2y^2+4y+7$
I am not able to solve this equation though i am able to differentiate it. 

For $x^2+axy+bx+cy-d=0,$
$$x^2+x(ay+b)+cy-d=0$$
As $x$ is real, the discriminant must be $\ge0$ 
i.e.,  $$(ay+b)^2\ge4(cy-d)\iff4d\ge4cy-(a^2y^2+2aby+b^2)$$
$$=-b^2-a^2y^2-2y(ab-2c)$$
$$=\left(\dfrac{ab-2c}a\right)^2-b^2-\left(ay+\dfrac{ab-2c}a\right)^2$$
$$\ge\left(\dfrac{ab-2c}a\right)^2-b^2$$ 
the equality occurs if $ay+\dfrac{ab-2c}a=0$
