Question 1: Factorise the expression: 6x^3 + 9x^2.

Answer: 3x^2(2x + 3)

Question 2: Simplify by extracting the common factor: 8a^2b - 12ab^2.

Answer: 4ab(2a - 3b)

Question 3: Factorise completely: 15x^2y + 10xy^2.

Answer: 5xy(3x + 2y)

Question 4: Calculate the factorised form of 18p^2q - 24pq^2, then verify by expanding.

Answer: Factorised: 6pq(3p - 4q); expanded: 18p^2q - 24pq^2

Question 5: Identify the common factor and factorise: 14m^2n - 21mn^2 + 7m.

Answer: 7m(2mn - 3n^2 + 1)

Question 6: Solve for x: Factorise 4x^2 + 8x, then find the values of x.

Answer: x = 0 or x = -2

Question 7: Word problem: A rectangle has sides represented by algebraic expressions 3x + 2 and 6x + 4. Factorise the expression for the perimeter and find its form in terms of x.

Answer: Perimeter: 2(3x + 2 + 6x + 4) = 2(9x + 6) = 18x + 12

Question 8: Construct a quadratic expression involving a common factor of 5, and then factorise it.

Answer: Example: 5x^2 + 10x = 5x(x + 2)

Question 9: Challenge: Given the expression 12a^3b^2 - 8a^2b^3 + 4a^2b, factorise completely and explain each step.

Answer: Factor out 4a^2b: 4a^2b(3a - 2b + 1)

Question 10: Error analysis: The expression 9x^2 + 15x is incorrectly factored as 3x(3x + 5). Identify and correct the mistake.

Answer: Incorrect: 3x(3x + 5). Correct factorisation: 3x(3x + 5) is correct; the mistake might be assuming common factor for 9x^2 + 15x, which actually is 3x(3x + 5).