<s>[INST] Show that this expression is a perfect square? Show that this expression is a perfect square?
$(b^2 + 3a^2 )^2 - 4 ab*(2b^2 - ab - 6a^2)$
 [/INST] Once you factored it out and got $9a^4+24a^3b+10a^2b^2−8ab^3+b^4$, you'd suspect that this is the square of a sum. What are the parts of the sum? 
If we let b = 0, the sum is $9a^4$ so there should be a term $±3a^2$. If we let a = 0, the sum is $b^4$, so there should be a term $±b^2$. 
We know that $(x+y)^2=x^2 + 2xy + y^2$. Guessing that $x=3a^2$ and $24a^3b=2xy=6a^2y$, that would make y = 4ab. From the other side, if $x=±b^2$ and $2xy=-8ab^3=-/+2yb^2$, that makes $x = -b^2$ and $y = 4ab$. 
So an educated guess would be $(3a^2+4ab-b^2)^2$, which lucky enough produces the requested result. 
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