quadratic

\hspace{-16}$Let $\mathbf{x^4+ax^3+bx^2+cx+d =\left(x^2+px+q\right)^2}$\\\\\\ $\mathbf{x^4+ax^3+bx^2+cx+d=x^4+2px^3+(p^2+2q).x^2+2pq.x+q^2}$\\\\ Now equating Coeff. of $\mathbf{x^3\;,x^2\;,x}$ and Constant term, we Get\\\\ $\mathbf{\begin{Bmatrix} \bold{a=2p} \\\\ \bold{b=p^2+2q} \\\\ \bold{c=2pq} \\\\ \bold{d=q^2} \end{Bmatrix}}$\\\\\\ So from here $\mathbf{c^2=4p^2.q^2=(2p)^2.q^2=a^2.d^2}$