<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] In fact, note that $(x + y - z)^2 = x^2 + y^2 + z^2 + 2xy - 2yz - 2xz$. Thus,
$$
(3a^2 + b^2)^2 - 8ab^3 + 4a^2b^2 + 24a^3b = 9a^4 + 10a^2b^2 + b^4 - 8ab^3 + 24a^3b
$$
$$
= 9a^4 + (16a^2b^2 - 6a^2b^2) + b^4 + 24a^3b - 8ab^3 
$$
$$
= 9a^4 + 16a^2b^2 + b^4 + 24a^3b - 6a^2b^2 - 8ab^3
$$
$$
= (3a^2)^2 + (4ab)^2 + (-b^2)^2 + 2(3a^2)(4ab) + 2(3a^2)(-b^2) + 2(4ab)(-b^2) 
$$
$$
= (3a^2 + 4ab - b^2)^2
$$
</s>