Question 1: Prove that (a + b)^2 = a^2 + 2ab + b^2 for any real numbers a and b.

Answer: Proved by expanding (a + b)^2 = (a + b)(a + b) = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2.

Question 2: Show that the difference of squares identity holds: a^2 - b^2 = (a + b)(a - b).

Answer: Proved by expanding (a + b)(a - b) = a^2 - ab + ab - b^2 = a^2 - b^2.

Question 3: Verify that 3(x + 2) = 3x + 6 for any value of x.

Answer: Distribute 3: 3x + 6, which matches the right side.

Question 4: Show that the identity 5(2y - 3) = 10y - 15 is true for any y.

Answer: Distribute 5: 10y - 15, matching the right side.

Question 5: Construct a proof to verify that (x + 1)^2 = x^2 + 2x + 1.

Answer: Expand (x + 1)^2 = (x + 1)(x + 1) = x^2 + x + x + 1 = x^2 + 2x + 1.

Question 6: Prove that the identity (a + b)^2 = a^2 + 2ab + b^2 holds when a=3 and b=4.

Answer: Substitute: (3 + 4)^2 = 7^2 = 49; RHS: 3^2 + 2*3*4 + 4^2 = 9 + 24 + 16 = 49. Identity holds.

Question 7: Prove that (x - y)^2 = x^2 - 2xy + y^2.

Answer: Expand: (x - y)^2 = (x - y)(x - y) = x^2 - xy - xy + y^2 = x^2 - 2xy + y^2.

Question 8: Simplify the expression: 2(a + b) + 3(a - b). Prove the result matches 5a - b.

Answer: Distribute: 2a + 2b + 3a - 3b = (2a + 3a) + (2b - 3b) = 5a - b.

Question 9: A student claims that (x + 2)^2 = x^2 + 4x + 4. Is this correct? Prove or disprove.

Answer: Correct. Expand: (x + 2)^2 = (x + 2)(x + 2) = x^2 + 2x + 2x + 4 = x^2 + 4x + 4.

Question 10: Identify the mistake in this proof: The student claims that (a + b)^2 = a^2 + b^2 because they expanded (a + b)^2 as a^2 + b^2.

Answer: Mistake: They ignored the middle term 2ab. Correct expansion is a^2 + 2ab + b^2.

Question 11: Prove that the identity (x + y)(x - y) = x^2 - y^2 holds, using algebraic expansion.

Answer: Expand: (x + y)(x - y) = x^2 - xy + xy - y^2 = x^2 - y^2.