Question 1: Expand (x + 2 + y)^3 using step-by-step expansion.

Answer: x^3 + 6x^2 + 12x + 8 + 3x^2y + 6xy + 3y^2 + 3x y^2 + 6 y + y^3

Question 2: Calculate (a + b + 1)^3 and show each step.

Answer: a^3 + 3a^2 + 3a + 1 + 3a^2b + 6ab + 3b^2 + 3a b^2 + 6b + 1

Question 3: Expand (3 + x + y)^3 in steps.

Answer: 27 + 27x + 27y + 9x^2 + 18xy + 9y^2 + x^3 + 3x^2 y + 3x y^2 + y^3

Question 4: Simplify the expansion of (x + 3 + y)^3 fully.

Answer: x^3 + 9x^2 + 27x + 27 + 3x^2 y + 9xy + 3y^2 + 3x y^2 + 9 y + y^3

Question 1: Expand (x + y + 4)^3 and identify the coefficient of x^2 y.

Answer: 18

Question 2: Given (a + 2 + b)^3, find the coefficient of a^2 b and explain your steps.

Answer: Coefficient is 3

Question 3: If (x + 1 + y)^3 = x^3 + 3x^2 + 3x + 1 + 3x^2 y + 6xy + 3 y^2 + 3 x y^2 + 6 y + y^3, verify the expansion by re-deriving the terms.

Answer: Verified step-by-step expansion matches given expression.

Question 4: Construct a mini-proof to show that expanding (a + b + c)^3 always results in 1 + 3a + 3b + 3c + 3a^2 + 3b^2 + 3c^2 + 6ab + 6 ac + 6 bc + a^3 + b^3 + c^3, using step-by-step expansion.

Answer: Step-by-step expansion confirms the pattern.

Question 1: A box is being assembled with dimensions described by (x + 2 + y). If the volume of the box is given by cubing this expression, expand to find an expression for volume.

Answer: x^3 + 6x^2 + 12x + 8 + 3x^2 y + 6xy + 3y^2 + 3x y^2 + 6 y + y^3

Question 2: A garden has sides described by (a + b + 1). If you want to find the total area when squared (area = (a + b + 1)^2), expand and interpret the result.

Answer: a^2 + 2a + 1 + 2ab + 2b + b^2

Question 1: Prove that (x + y + z)^3 expands to the sum of all cubic terms with coefficients as per binomial expansion pattern, using step-by-step expansion.

Answer: All cubic terms with correct coefficients derived as per expansion.

Question 2: Given (p + q + r)^3, identify the term with coefficient 6 in the expansion and explain how it arises.

Answer: The term 6 p q r arises from the expansion coefficient for the mixed term pq r.

Question 1: A student expands (x + y + 2)^3 but forgets to include the term 3x y^2. Identify the mistake and correct the expansion.

Answer: The missing term is 3x y^2; the corrected expansion adds this term.

Question 2: A common error is to miscalculate the coefficient for the term x^2 y. Describe how to correctly determine this coefficient.

Answer: Coefficient is 3, derived from the multinomial coefficient for the term with x^2 y.