1. **State the problem:** Simplify the expression $2x \times 2x^2$. 2. **Recall the multiplication rule for exponents:** When multiplying terms with the same base, add the exponents: $x^a \times x^b = x^{a+b}$. 3. **Apply the rule:** $$2x \times 2x^2 = (2 \times 2) \times (x^1 \times x^2)$$ 4. **Simplify coefficients:** $$2 \times 2 = 4$$ 5. **Add exponents of $x$:** $$x^{1+2} = x^3$$ 6. **Final simplified expression:** $$4x^3$$ So, $2x \times 2x^2 = 4x^3$.