1. **State the problem:** Solve the equation $$1.1 \times 10^{-12} - 2x^3 - 0.2x^2 = 0$$ for $x$. 2. **Rewrite the equation:** Move all terms except the constant to the right side: $$1.1 \times 10^{-12} = 2x^3 + 0.2x^2$$ 3. **Factor the right side:** Factor out $x^2$: $$1.1 \times 10^{-12} = x^2(2x + 0.2)$$ 4. **Isolate terms:** We want to find $x$ such that: $$x^2(2x + 0.2) = 1.1 \times 10^{-12}$$ 5. **Check for possible roots:** Since the right side is very small, consider if $x$ is very small. Try to solve the cubic equation: $$2x^3 + 0.2x^2 - 1.1 \times 10^{-12} = 0$$ 6. **Use substitution or numerical methods:** This cubic is not easily factorable by hand. However, note that for very small $x$, $2x^3$ is negligible compared to $0.2x^2$. Approximate: $$0.2x^2 \approx 1.1 \times 10^{-12}$$ 7. **Solve for $x^2$:** $$x^2 = \frac{1.1 \times 10^{-12}}{0.2} = 5.5 \times 10^{-12}$$ 8. **Find $x$:** $$x = \pm \sqrt{5.5 \times 10^{-12}} = \pm 2.345 \times 10^{-6}$$ 9. **Check the approximation:** Substitute $x = 2.345 \times 10^{-6}$ back into the original equation to verify the solution is close. **Final answer:** $$x \approx \pm 2.345 \times 10^{-6}$$