Cauchy Schwarz: Proving this inequality For positive reals a,b,x and y, prove that 
$(5a^2 + 2ab + 3b^2)(5x^2 + 2xy + 3y^2) \ge (5ax + ay + bx + 3by)^2$
My attempt: By Cauchy we can show that the LHS $\ge(5ax + 2(abxy)^{1/2} + 3by)^2$
I can then show that $ay + bx \ge 2(abxy)^{1/2}$ using AM-GM, but this doesn't help.
How can I proceed?
Thank you.

By C-S we obtain:
$$(5a^2+2ab+3b^2)(5x^2+2xy+3y^2)=\left(\frac{1}{5}(5a+b)^2+\frac{14}{5}b^2\right)\left(\frac{1}{5}(5x+y)^2+\frac{14}{5}y^2\right)\geq$$
$$\geq\left(\frac{1}{5}(5a+b)(5x+y)+\frac{14}{5}by\right)^2=(5ax+ay+bx+3by)^2.$$
