In basic algebra, what is the simplified result when you add x squared and x squared, that is, when you evaluate x^2 + x^2?

Introduction / Context:
This question checks a very fundamental concept in algebra: combining like terms. The expression "x squared plus x squared" means x^2 + x^2. Both terms have exactly the same variable part and the same power, so they are like terms and can be added in a straightforward way by combining their coefficients.

Given Data / Assumptions:

• The expression is x^2 + x^2.
• x is an algebraic variable and can represent any real number.
• We use standard rules of algebra for addition of like terms.

Concept / Approach:
When adding or subtracting like terms in algebra, we keep the common variable part the same and add or subtract only the numerical coefficients. For example, 3x + 5x = (3 + 5)x = 8x. In the same way, x^2 can be viewed as 1x^2, so x^2 + x^2 = 1x^2 + 1x^2 = 2x^2. We do not change the exponent when adding like terms; powers change only when we multiply or divide terms with the same base.

Step-by-Step Solution:
Step 1: Rewrite each term with an explicit coefficient: x^2 = 1x^2. Step 2: So x^2 + x^2 = 1x^2 + 1x^2. Step 3: Add the coefficients: 1 + 1 = 2. Step 4: Keep the variable part exactly the same: x^2. Step 5: Therefore, x^2 + x^2 = 2x^2.

Verification / Alternative check:
We can verify using a trial value for x. Take x = 3. Then x^2 + x^2 = 9 + 9 = 18. On the other hand, 2x^2 = 2 * 9 = 18. Both sides match for this value of x, which supports the simplification. The same will happen for any other value of x, because the simplification is algebraically valid for all real x.

Why Other Options Are Wrong:
4x is obtained if someone wrongly treats x^2 as x and simply adds x + x + x + x. This ignores the square. x^4 comes from the mistaken idea that adding powers means adding exponents, which only applies to multiplication of like bases, not to addition. 2x is also incorrect because it again treats x^2 as x. The option x^2 ignores the fact that two equal terms are being added and therefore misses the factor 2.

Common Pitfalls:
A common confusion is between adding expressions and multiplying them. Students sometimes think that x^2 + x^2 should be x^4 or 2x, because they focus on the exponents instead of the coefficients. Always remember: when adding like terms, keep the variable part and exponent unchanged and add only the numerical coefficients in front of the term.

Final Answer:
Thus, the simplified form of x^2 + x^2, or "x squared plus x squared", is 2x^2.