1. The problem involves solving and simplifying a system of polynomial equations with variables $p$, $q$, $r$, $s$, $t$, and $a$, focusing on cubic equations and their relationships. 2. We start with the given cubic equation: $$s^3 + p s^2 + (p^2 - q^2)s - q^2 = 0$$ 3. The substitutions and relations given are: - $b = -p s$ - $q^2 = 6b + s^2$ - $b = 6a$ - $c = -\frac{1}{5}(q^2 - r)$ - $b = -p + t$ 4. The next step is to analyze the cubic in $t$: $$t^3 - p t^2 - 4 q^2 t + 8 p q^2 = \sqrt{t^3 - p t^2 - 4 q^2 t + 8 p q^2}$$ 5. To proceed, square both sides to eliminate the square root: $$\left(t^3 - p t^2 - 4 q^2 t + 8 p q^2\right)^2 = t^3 - p t^2 - 4 q^2 t + 8 p q^2$$ 6. This leads to a polynomial equation in $t$ which can be expanded and simplified to find roots or further relations. 7. The key is to isolate $t$ and solve the resulting polynomial, which may involve factoring or using the cubic formula. 8. This step is crucial for understanding the behavior of the system and finding explicit values for $t$, which then relate back to $b$, $s$, and $q^2$. Final answer: The next step is to square both sides of the equation involving $t$ to remove the square root and then solve the resulting polynomial: $$\left(t^3 - p t^2 - 4 q^2 t + 8 p q^2\right)^2 = t^3 - p t^2 - 4 q^2 t + 8 p q^2$$