What is the simplified expanded form of (x + 3)^2 + (x − 1)^2?

Introduction / Context:
This is a straightforward algebraic expansion and simplification question. It tests your ability to expand binomials using the formula (a + b)^2 = a^2 + 2ab + b^2 and then combine like terms efficiently.

Given Data / Assumptions:

- Expression: (x + 3)^2 + (x − 1)^2
- x is a real variable
- We want the simplified polynomial form of the given expression

Concept / Approach:
The approach is to expand each squared binomial separately, then add the results and collect like terms in x^2, x, and constant form. Doing this methodically prevents sign errors and helps arrive at a clean simplified expression.

Step-by-Step Solution:
Step 1: Expand (x + 3)^2 using the identity (a + b)^2 = a^2 + 2ab + b^2.Step 2: (x + 3)^2 = x^2 + 2 * x * 3 + 3^2 = x^2 + 6x + 9.Step 3: Expand (x − 1)^2 similarly: (x − 1)^2 = x^2 − 2 * x * 1 + 1^2 = x^2 − 2x + 1.Step 4: Add the two expanded expressions: (x^2 + 6x + 9) + (x^2 − 2x + 1).Step 5: Combine like terms: x^2 + x^2 = 2x^2; 6x − 2x = 4x; 9 + 1 = 10. So the sum is 2x^2 + 4x + 10.Step 6: Factor out 2 to match one of the answer choices: 2x^2 + 4x + 10 = 2(x^2 + 2x + 5).

Verification / Alternative check:
You can check this by substituting a simple value for x, such as x = 1. The original expression gives (1 + 3)^2 + (1 − 1)^2 = 4^2 + 0^2 = 16. The simplified expression 2(x^2 + 2x + 5) at x = 1 gives 2(1 + 2 + 5) = 2 * 8 = 16. The results match, confirming the simplification.

Why Other Options Are Wrong:
- x^2 + 2x + 5 is only half of the correct expression and does not match numerical checks.
- x^2 − 2x + 5 and 2(x^2 − 2x + 5) arise from sign mistakes in the expansion of (x − 1)^2 or from forgetting to add the coefficients correctly.
- x^2 + 4x + 10 represents 2x^2 + 4x + 10 divided by 2 incorrectly, not the full simplified sum.

Common Pitfalls:
Students often make sign errors when expanding (x − 1)^2 or forget to square the constant term. Another mistake is not combining like terms correctly. Writing each step clearly and using the standard binomial formula helps avoid these small but costly errors.

Final Answer:
The simplified form is 2(x^2 + 2x + 5).