We have to prove that x^2 + 2xy + 3y^2 - 6x - 2y > -11

A term in an expression that is a square cannot be negative, therefore:

(x + 2y)^2 + (x - 6)^2 + 2(y - 1)^2 + 38 `>=` 38

An expression greater than 38 is greater than 0

=> (x + 2y)^2 + (x - 6)^2 + 2(y - 1)^2 + 38 `>=` 0

=> x^2 + 4xy + 4y^2 + x^2 - 12x + 36 + 2y^2 - 4y + 2 - 38 `>=` 0

=> 2x^2 + 4xy + 6y^2 - 12x - 4y - 38 `>=` 0

=> x^2 + 2xy + 3y^2 - 6x - 2y - 19 `>=` 0

=> x^2 + 2xy + 3y^2 - 6x - 2y `>=` 19

If an expression is greater than 19 it is greater than -11.

This proves that x^2 + 2xy + 3y^2 - 6x - 2y `>=` -11