1. Problem 43: A student simplified $ (x + 5)^2 $ as $ x^2 + 25 $. Identify the error. 2. Recall the expansion formula for a binomial square: $$ (a + b)^2 = a^2 + 2ab + b^2 $$ 3. Applying this to $ (x + 5)^2 $, we get: $$ x^2 + 2 \times x \times 5 + 5^2 = x^2 + 10x + 25 $$ 4. The student wrote $ x^2 + 25 $ and missed the middle term $ 10x $. 5. Therefore, the error is that the student forgot to include the middle term $ 10x $. --- 6. Problem 44: Check the solution $ x^2 - 16 = (x + 8)(x - 8) $ for errors. 7. Recognize that $ x^2 - 16 $ is a difference of squares: $$ a^2 - b^2 = (a + b)(a - b) $$ where $ a = x $ and $ b = 4 $ because $ 16 = 4^2 $. 8. The correct factorization is: $$ x^2 - 16 = (x + 4)(x - 4) $$ 9. The solution given uses $ 8 $ instead of $ 4 $, so the constant term is incorrect. 10. The error is that the factoring method used was a difference of two squares, but the factors are incorrect. --- 11. Problem 45: A square with area $ 25u^2 $ has a smaller square cut out with area $ 16 $. Find the remaining area. 12. The side length of the large square is: $$ \sqrt{25u^2} = 5u $$ 13. The side length of the smaller square is: $$ \sqrt{16} = 4 $$ 14. The remaining area is the difference of squares: $$ (5u)^2 - 4^2 = (5u - 4)(5u + 4) $$ 15. Therefore, the expression representing the remaining area is: $$ (5u + 4)(5u - 4) $$ --- Final answers: 43. A. The student forgot to include the middle term, $10x$. 44. A. The factoring method used was a difference of two squares, but the factors are incorrect. 45. B. The remaining area is represented by $ (5u + 4)(5u - 4) $.