PROVE THAT X^2+2XY+3Y^2-6X-2Y CANNOT BE LESS THAN -11. X,Y BELONG TO ALL REAL NOS.....THANKS FOR HELP!

Equivalently we have to prove that 2x²+4xy+6y²-12x-4y+22>=0

From Cauchy Schwarz (CS) Inequality, 4 [(x+2y)²+(y-1)²+(y-1)²+(x-6)²] >= (x+2y+1-y+1-y+6-x)² = 64

Hence, (x+2y)²+(y-1)²+(y-1)²+(x-6)² >= 16

or 2x²+4xy+6y²-12x-4y+22>=0

Since the CS equality condition cannot be met here, we have 2x²+4xy+6y²-12x-4y+22>0