If $\alpha$ and $\beta$ are the roots of $x^2+px+q=0$ and $\alpha^4, \beta^4$ are the roots of $x^2-rx+5=0$, then the equation $x^2-4qx+2q^2-r=0$ has always

Given the roots of x^2 + px + q = 0 are alpha and beta, alpha^4 and beta^4 are roots of x^2 - rx + 5 = 0. Using symmetric properties, the product of roots alpha^4 * beta^4 = q^4 = 5. The equation x^2 - 4qx + 2q^2 - r = 0 has a discriminant D = 16q^2 - 4(2q^2 - r) = 8q^2 + 4r. Since q^4 = 5, q^2 = sqrt(5). The constant term is 2q^2 - r. Analysis shows the product of roots is negative, implying one positive and one negative root.