1. **State the problem:** Solve the complex equation $$p(z) = z^2 - 3z^2 + z + 5$$ for $z$. 2. **Simplify the polynomial:** Combine like terms: $$p(z) = z^2 - 3z^2 + z + 5 = -2z^2 + z + 5$$ 3. **Set the polynomial equal to zero to find roots:** $$-2z^2 + z + 5 = 0$$ 4. **Rewrite the equation:** $$-2z^2 + z + 5 = 0 \implies 2z^2 - z - 5 = 0$$ 5. **Use the quadratic formula:** For $az^2 + bz + c = 0$, roots are $$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=2$, $b=-1$, $c=-5$. 6. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-1)^2 - 4(2)(-5) = 1 + 40 = 41$$ 7. **Find the roots:** $$z = \frac{-(-1) \pm \sqrt{41}}{2 \times 2} = \frac{1 \pm \sqrt{41}}{4}$$ 8. **Final answer:** The solutions to the equation are $$z = \frac{1 + \sqrt{41}}{4} \quad \text{and} \quad z = \frac{1 - \sqrt{41}}{4}$$