Can you figure out the easy way to solve this

Suppose x^2 + 3y^2 = 52 and x y = 12

Find the value of 4x^4 - 4x^2 y^2 + 9y^4

There is an "obvious" way that will get the answer pretty easily. But there is a less obvious way that will get it even easier. I'm just curious to see if anyone can find it.

I guess it's (2x^2 + 3y^2 + 4xy)(2x^2 + 3y^2 - 4xy)

= (52 + 48)(52 - 48)
= 100 * 4
= 400

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Sorry ... didn't explain that particularly well :(

(2x^2 + 3y^2)^2 = 4x^4 + 12x^2y^2 + 9y^4

We need to subtract (4xy)^2 to get to the target number.

Difference between two squares 'n' all that.

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@Marc: I think it's 4 * 576 that needs to be subtracted.

First, start with the initial equation:
2x^2 + 3y^2 = 52

Square the equation:
4x^2 + 12x^2 y^2 + 9y^4 = 2704

You are trying to solve:
4x^4 - 4x^2 y^2 + 9y^4

To get from the first equation to the second, subtract 16x^2 y^2:
4x^2 + 12x^2 y^2 - 16x^2 y^2 + 9y^4 = 2704 - 16x^2 y^2
4x^4 - 4x^2 y^2 + 9y^4 = 2304 - 16x^2 y^2

xy = 12
(xy)^2 = 12^2
x^2 y^2 = 144

Multiplying each side by 16 gets you:
16x^2 y^2 = 144*16
16x^2 y^2 = 2304

Using our predetermined equation, we can deduce the following:
4x^4 - 4x^2 y^2 + 9y^4 = 2304 - 16x^2 y^2
4x^4 - 4x^2 y^2 + 9y^4 = 2704 - 2304
4x^4 - 4x^2 y^2 + 9y^4 = 400

if xy=12, then -4x^2y^2=-576
if 2x^2+3y^2=52, then 4x^4 +9y^4= 52 * 52 =2704
2704 - 576 =2128.

COOL!

Square the first equation, square the second and times by 4 then the answer to the third equation will be
(52)^2-4(12)^2