Question 1: Factorise the expression 6x + 9.

Answer: 3(2x + 3)

Question 2: Simplify the algebraic fraction (4x^2 + 8x) / (2x).

Answer: 2x + 4

Question 3: Factorise 12y^2 - 18y.

Answer: 6y(2y - 3)

Question 4: Express the fraction (9a^2 + 3a) / (3a).

Answer: 3a + 1

Question 5: In a rectangle, the length is represented as 4(3x + 2), and the width as 2(2x + 1). Factorise these expressions first to find a simplified ratio of length to width.

Answer: Length : Width = (12x + 8) / (4x + 2), which simplifies to 3:1 after factorising.

Question 6: Factorise the expression 15z + 20.

Answer: 5(3z + 4)

Question 7: Simplify the fraction (6x^2 + 9x) / (3x).

Answer: 2x + 3

Question 8: Factorise 8m^2 - 12m + 4.

Answer: 4(2m^2 - 3m + 1)

Question 9: Determine the factorised form of the expression 20x^2 + 25x.

Answer: 5x(4x + 5)

Question 10: A bag contains 3(2x + 4) apples and 2(3x + 2) oranges. Factorise these expressions to compare their quantities.

Answer: Apples: 6x + 12, Oranges: 6x + 4; after factorising, apples are 3(2x + 4), oranges are 2(3x + 2).

Question 11: Identify and correct the mistake: Factorising 4x + 5y yields 2(2x + 5y).

Answer: Incorrect. 4x + 5y cannot be factorised further as 2(2x + 2.5y). The original expression has no common factor.

Question 12: Challenge: Factorise 18x^2 + 24x + 6 and verify the factorisation.

Answer: 6(3x^2 + 4x + 1); verify by expanding.