1. **State the problem:** Solve the equation $2x^3 + 0.10x = 4.0 \times 10^{-11}$ for $x$. 2. **Rewrite the equation:** $$2x^3 + 0.10x = 4.0 \times 10^{-11}$$ 3. **Isolate terms:** We want to find $x$ such that the equation holds. 4. **Factor out $x$ on the left side:** $$x(2x^2 + 0.10) = 4.0 \times 10^{-11}$$ 5. **Since the right side is very small, consider the possibility that $x$ is very small.** 6. **Try to solve numerically or approximate:** Because this is a cubic equation, exact algebraic solution is complex. We can try to approximate. 7. **Check if $x$ is very small, then $2x^3$ is negligible compared to $0.10x$:** Approximate: $$0.10x \approx 4.0 \times 10^{-11} \implies x \approx \frac{4.0 \times 10^{-11}}{0.10} = 4.0 \times 10^{-10}$$ 8. **Check if this $x$ satisfies the original equation:** Calculate $2x^3 = 2(4.0 \times 10^{-10})^3 = 2(64 \times 10^{-30}) = 1.28 \times 10^{-28}$, which is negligible compared to $0.10x = 4.0 \times 10^{-11}$. 9. **Therefore, the approximate solution is:** $$\boxed{x \approx 4.0 \times 10^{-10}}$$