1. **State the problem:** Solve for $x$ in the quadratic equation $$-x^2 + 7x - 19 = 0.$$\n\n2. **Rewrite the equation:** Multiply both sides by $-1$ to make the leading coefficient positive:\n$$\cancel{-1}(-x^2 + 7x - 19) = \cancel{-1}(0) \implies x^2 - 7x + 19 = 0.$$\n\n3. **Identify coefficients:** Here, $a = 1$, $b = -7$, and $c = 19$.\n\n4. **Use the quadratic formula:** For $ax^2 + bx + c = 0$, solutions are given by\n$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$\n\n5. **Calculate the discriminant:**\n$$\Delta = b^2 - 4ac = (-7)^2 - 4(1)(19) = 49 - 76 = -27.$$\n\n6. **Interpret the discriminant:** Since $\Delta < 0$, there are no real solutions; solutions are complex.\n\n7. **Find the complex solutions:**\n$$x = \frac{-(-7) \pm \sqrt{-27}}{2(1)} = \frac{7 \pm \sqrt{-27}}{2} = \frac{7 \pm \sqrt{27}i}{2}.$$\n\n8. **Simplify $\sqrt{27}$:**\n$$\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}.$$\n\n9. **Final solutions:**\n$$x = \frac{7 \pm 3\sqrt{3}i}{2}.$$