Question 1
Show that real numbers $p$, $q$, $r$ satisfy the condition
$$
p^4(q-r)^2 + 2p^2(q+r) + 1 = p^4
$$
if and only if the quadratic equations
$$
x^2 + px + q = 0 \quad \text{and} \quad y^2 - py + r = 0
$$
have real roots (not necessarily distinct) which can be labelled by $x_{1,2}$ and $y_{1,2}$, respectively, in such way that the equality $x_1y_1 - x_2y_2 = 1$ holds.
$$
p^4(q-r)^2 + 2p^2(q+r) + 1 = p^4
$$
if and only if the quadratic equations
$$
x^2 + px + q = 0 \quad \text{and} \quad y^2 - py + r = 0
$$
have real roots (not necessarily distinct) which can be labelled by $x_{1,2}$ and $y_{1,2}$, respectively, in such way that the equality $x_1y_1 - x_2y_2 = 1$ holds.