Linear combinations of squares

How many pairs of integers (not necessarily positive) are there such that both a2+6b2a^2+6b^2 and b2+6a2b^2 + 6a^2 are both squares?

This problem is proposed by Shivang.

Details and assumptions

You may choose to read the post on Gaussian Integers.

You may submit original proposed problems to Calvin@Brilliant.org.

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