1. **State the problem:** Solve for $x$ in the equation $$4.8 \times 10^3 = 2x^2 (0.0100 - x)(0.0200 + x).$$ 2. **Rewrite the equation:** $$4.8 \times 10^3 = 2x^2 (0.0100 - x)(0.0200 + x).$$ 3. **Expand the product:** Use the distributive property for $(0.0100 - x)(0.0200 + x)$: $$ (0.0100)(0.0200) + (0.0100)(x) - x(0.0200) - x^2 = 0.0002 + 0.0100x - 0.0200x - x^2 = 0.0002 - 0.0100x - x^2. $$ 4. **Substitute back:** $$4.8 \times 10^3 = 2x^2 (0.0002 - 0.0100x - x^2).$$ 5. **Distribute $2x^2$:** $$4.8 \times 10^3 = 2x^2 \times 0.0002 - 2x^2 \times 0.0100x - 2x^2 \times x^2 = 0.0004x^2 - 0.0200x^3 - 2x^4.$$ 6. **Rewrite as a polynomial equation:** $$0 = 0.0004x^2 - 0.0200x^3 - 2x^4 - 4.8 \times 10^3.$$ 7. **Rearranged:** $$-2x^4 - 0.0200x^3 + 0.0004x^2 - 4800 = 0.$$ 8. **Multiply both sides by $-1$ to simplify signs:** $$2x^4 + 0.0200x^3 - 0.0004x^2 + 4800 = 0.$$ 9. **This is a quartic equation in $x$.** Solving analytically is complex; numerical methods or graphing are recommended. 10. **Summary:** The equation reduces to $$2x^4 + 0.0200x^3 - 0.0004x^2 + 4800 = 0,$$ which can be solved numerically for $x$. **Final answer:** The solution(s) for $x$ satisfy $$2x^4 + 0.0200x^3 - 0.0004x^2 + 4800 = 0.$$