Four roots of the polynomial $x^4+px^3​+qx^2​+rx+1$ Let $a, b, c, d$ be four roots of the polynomial $x^4+px^3​+qx^2​+rx+1$ prove that $(a^4​+1)(b^4​+1)(c^4​+1)(d^4​+1)=(p^2​+r^2)​^2​+q^4-4pq^2​r$
Please provide hint. 

$$\prod_{cyc}(a^4+1)=((a^2b^2-1)^2+(a^2+b^2)^2)((c^2d^2-1)^2+(c^2+d^2)^2)=$$
$$=((a^2b^2-1)(c^2+d^2)+(c^2d^2-1)(a^2+b^2))^2+$$
$$+((a^2b^2-1)(c^2d^2-1)-(a^2+b^2)(c^2+d^2))^2=$$
$$=\left(\sum_{cyc}a^2b^2c^2-\sum_{cyc}a^2\right)^2+\left(a^2b^2c^2d^2-\frac{1}{6}\sum_{sym}a^2b^2\right)^2.$$
Can you end it now?
