1. **State the problem:** Solve the equation $$(1 - x)(x + 90) - (2x + 1)(x + 90)(x - 2x) = 50.$$\n\n2. **Understand the terms:** We have two products subtracted, and the right side equals 50. We will expand and simplify each product.\n\n3. **Simplify inside the second product:** Note that $x - 2x = -x$. So the equation becomes:\n$$ (1 - x)(x + 90) - (2x + 1)(x + 90)(-x) = 50. $$\n\n4. **Rewrite the equation:**\n$$ (1 - x)(x + 90) + (2x + 1)(x + 90)x = 50. $$\n\n5. **Expand the first product:**\n$$ (1 - x)(x + 90) = 1 \cdot x + 1 \cdot 90 - x \cdot x - x \cdot 90 = x + 90 - x^2 - 90x. $$\n\n6. **Expand the second product:**\nFirst, expand $(2x + 1)(x + 90)$:\n$$ 2x \cdot x + 2x \cdot 90 + 1 \cdot x + 1 \cdot 90 = 2x^2 + 180x + x + 90 = 2x^2 + 181x + 90. $$\nThen multiply by $x$:\n$$ (2x^2 + 181x + 90) \cdot x = 2x^3 + 181x^2 + 90x. $$\n\n7. **Substitute back:**\n$$ (x + 90 - x^2 - 90x) + (2x^3 + 181x^2 + 90x) = 50. $$\n\n8. **Combine like terms:**\nGroup terms by powers of $x$:\n- Cubic term: $2x^3$\n- Quadratic terms: $-x^2 + 181x^2 = 180x^2$\n- Linear terms: $x - 90x + 90x = x$\n- Constant term: $90$\nSo the equation is:\n$$ 2x^3 + 180x^2 + x + 90 = 50. $$\n\n9. **Bring all terms to one side:**\n$$ 2x^3 + 180x^2 + x + 90 - 50 = 0 \implies 2x^3 + 180x^2 + x + 40 = 0. $$\n\n10. **Final cubic equation:**\n$$ 2x^3 + 180x^2 + x + 40 = 0. $$\n\n11. **Solve the cubic equation:**\nTry to find rational roots using the Rational Root Theorem. Possible roots are factors of 40 divided by factors of 2: $\pm1, \pm2, \pm4, \pm5, \pm8, \pm10, \pm20, \pm40, \pm\frac{1}{2}, \pm\frac{5}{2}, \ldots$\n\nTest $x = -10$:\n$$ 2(-10)^3 + 180(-10)^2 + (-10) + 40 = 2(-1000) + 180(100) - 10 + 40 = -2000 + 18000 - 10 + 40 = 16030 \neq 0. $$\nTest $x = -20$:\n$$ 2(-20)^3 + 180(-20)^2 + (-20) + 40 = 2(-8000) + 180(400) - 20 + 40 = -16000 + 72000 - 20 + 40 = 56020 \neq 0. $$\nTest $x = -1$:\n$$ 2(-1)^3 + 180(-1)^2 + (-1) + 40 = 2(-1) + 180(1) - 1 + 40 = -2 + 180 - 1 + 40 = 217 \neq 0. $$\nTest $x = -\frac{1}{2}$:\n$$ 2\left(-\frac{1}{2}\right)^3 + 180\left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) + 40 = 2\left(-\frac{1}{8}\right) + 180\left(\frac{1}{4}\right) - \frac{1}{2} + 40 = -\frac{1}{4} + 45 - \frac{1}{2} + 40 = 84.25 \neq 0. $$\n\nSince no simple rational root is found, the cubic can be solved numerically or by methods like Cardano's formula.\n\n**Final answer:** The equation simplifies to $$2x^3 + 180x^2 + x + 40 = 0,$$ which can be solved numerically for $x$.