1. **State the problem:** Simplify and understand the expression and polynomial expansions given. 2. **Recall the formula for binomial expansion:** $$ (x + y)^2 = x^2 + 2xy + y^2 $$ This formula expands the square of a sum into three terms. 3. **Apply the formula to the expression:** Given $$ C = (x + y)^2 $$ Expanding using the formula: $$ C = x^2 + 2xy + y^2 $$ 4. **Check the provided expression:** The user wrote: $$ C = (x + y)^2 = x^2 + 2xy (x + y) - x^2 + 8x + 16 $$ This seems to have an error or typo because the middle term is incomplete or incorrect. 5. **Clarify the correct expansion:** The correct expansion is: $$ (x + y)^2 = x^2 + 2xy + y^2 $$ If the user meant to write: $$ (x + 4)^2 = x^2 + 8x + 16 $$ which is a specific case where $$ y = 4 $$. 6. **Simplify the arithmetic expressions:** - For $$ 2 - 1 + 3 - 5 $$: $$ 2 - 1 = 1 $$ $$ 1 + 3 = 4 $$ $$ 4 - 5 = -1 $$ - For $$ 4 - 12 $$: $$ 4 - 12 = -8 $$ - For $$ 3 - 1 + 6 $$: $$ 3 - 1 = 2 $$ $$ 2 + 6 = 8 $$ - For $$ 5 - 3 + 15 $$: $$ 5 - 3 = 2 $$ $$ 2 + 15 = 17 $$ 7. **Summary:** - The binomial expansion formula is used to expand squares of sums. - Arithmetic expressions are simplified step-by-step. **Final answers:** $$ (x + y)^2 = x^2 + 2xy + y^2 $$ $$ 2 - 1 + 3 - 5 = -1 $$ $$ 4 - 12 = -8 $$ $$ 3 - 1 + 6 = 8 $$ $$ 5 - 3 + 15 = 17 $$