Find the conditions that $a^{2} x^{4} + b x^{3} + c x^{2} + d x + f^{2}$ may be a perfect square.

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Solution

Let $a^{2} x^{4} + b x^{3} + c x^{2} + d x + f^{2} = (p x^{2} + q)^{2} R . H . S = p^{2} x^{4} + q^{2} + 2 p q x^{2}$
compare cofficients:
$a = p; b = 0; c = 2 p q; d = 0; f = q \Rightarrow c = 2 a f; b = 0; d = 0$

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