1. **State the problem:** Solve for $x$ in the equation $$4.0 \times 10^{-11} = x(2x + 0.10)^2.$$\n\n2. **Rewrite the equation:** Note that $0,10$ is a comma decimal notation, so rewrite as $$4.0 \times 10^{-11} = x(2x + 0.10)^2.$$\n\n3. **Expand the squared term:** $$(2x + 0.10)^2 = (2x)^2 + 2 \times 2x \times 0.10 + (0.10)^2 = 4x^2 + 0.4x + 0.01.$$\n\n4. **Substitute back:** $$4.0 \times 10^{-11} = x(4x^2 + 0.4x + 0.01) = 4x^3 + 0.4x^2 + 0.01x.$$\n\n5. **Rewrite as a cubic equation:** $$4x^3 + 0.4x^2 + 0.01x - 4.0 \times 10^{-11} = 0.$$\n\n6. **Analyze the equation:** This is a cubic polynomial in $x$. Because the constant term is very small, and coefficients are small, we expect a very small root.\n\n7. **Approximate solution:** For very small $x$, $4x^3$ and $0.4x^2$ are negligible compared to $0.01x$, so approximate $$0.01x \approx 4.0 \times 10^{-11} \Rightarrow x \approx \frac{4.0 \times 10^{-11}}{0.01} = 4.0 \times 10^{-9}.$$\n\n8. **Check approximation:** Substitute $x = 4.0 \times 10^{-9}$ back into the cubic terms to verify they are negligible.\n\n**Final answer:** $$x \approx 4.0 \times 10^{-9}.$$