Question 1: Factorise the quadratic expression 3x² + 6x.

Answer: 3(x + 2)(x)

Question 2: Express 6x² + 11x + 3 as a product of two binomials.

Answer: (2x + 1)(3x + 3)

Question 3: Factorise the quadratic 4x² + 4x - 8 and simplify the expression.

Answer: 4(x + 2)(x - 1)

Question 4: Solve for x: 2x² + 7x + 3 = 0 by factorising.

Answer: x = -3/2 or x = -1/2

Question 5: Factorise the quadratic 5x² + 10x + 5 and state its common factor.

Answer: 5(x + 1)²

Question 6: Construct a quadratic expression with leading coefficient 3, a middle term of 7x, and a constant of 6, then factorise it.

Answer: 3x² + 7x + 6 = (3x + 2)(x + 3)

Question 7: A rectangular garden has an area represented by 2x² + 8x. Factorise the expression to find possible dimensions.

Answer: 2x(x + 4)

Question 8: Identify the common mistake in factorising 4x² + 5x + 1 as (4x + 1)(x + 1).

Answer: Incorrect middle term; it should be (4x + 1)(x + 1) which expands to 4x² + 5x + 1, so the mistake is in the expansion process (no mistake here). Alternatively, students should identify that the factorisation is correct, but if they see it as an error, they should correct a similar mistake.

Question 9: Challenge: Factorise 9x² + 24x + 16 and write it in the form (ax + b)².

Answer: (3x + 4)²

Question 10: Explain why the quadratic 2x² + 4x + 2 cannot be factorised into two distinct linear factors over the real numbers.

Answer: Because its discriminant is zero, indicating a perfect square trinomial, so it factors as 2(x + 1)².