1. **State the problem:** Simplify the expression $x \times x + x \times x$. 2. **Recall the rules:** Multiplying a variable by itself is squaring it, so $x \times x = x^2$. 3. **Apply the rule:** Replace each $x \times x$ with $x^2$. 4. **Simplify the expression:** $$x^2 + x^2 = 2x^2$$ 5. **Final answer:** The simplified form of the expression is $2x^2$.