1. The problem is to verify the transformation and simplification of the quartic equation $$x^4 + b x^3 + c x^2 + d x + e = 0$$ into the depressed quartic form $$y^4 + p y^2 + q y + r = 0$$ using the substitution $$x = y - \frac{b}{4a}$$. 2. The substitution is used to eliminate the cubic term in the quartic equation. The general quartic is $$a x^4 + b x^3 + c x^2 + d x + e = 0$$, and dividing by $$a$$ gives the monic form: $$x^4 + \frac{b}{a} x^3 + \frac{c}{a} x^2 + \frac{d}{a} x + \frac{e}{a} = 0$$. 3. Substituting $$x = y - \frac{b}{4a}$$ and expanding yields a quartic in $$y$$ without the cubic term: $$y^4 + p y^2 + q y + r = 0$$ where $$p = - \frac{3 b^2}{8 a^2} + \frac{c}{a}$$ $$q = \frac{b^3}{8 a^3} - \frac{b c}{2 a^2} + \frac{d}{a}$$ $$r = - \frac{3 b^4}{256 a^4} + \frac{b^2 c}{16 a^3} - \frac{b d}{4 a^2} + \frac{e}{a}$$. 4. The given expressions match the standard formulas for the depressed quartic coefficients after the substitution. 5. Therefore, the transformation and the expressions for $$p$$, $$q$$, and $$r$$ are correct and consistent with the theory of solving quartic equations by depressing the cubic term. Final answer: The check confirms the correctness of the substitution and the resulting depressed quartic form.