Hi Archit,

If $P(x)$ is a square it must be of the form $\left(c_1 x^2 + c_2 x + c_3\right)^2.$ Thus you have

\[x^4 + a x^3 + b x^2 - 8 x + 1 = \left(c_1 x^2 + c_2 x + c_3\right)^2.\]

Expand the right side and compare the coefficients on the right and left sides. What are the possible values of $c_1?$ What are the possible values of $c_3?$ Given these values what can $c_2$ be? Now that you know the possible values of the coefficients on the right side what can you say about $a \mbox{and} b?$

Penny