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

Answer: 3x(2x + 3)

Question 2: Simplify the algebraic fraction \(rac{4x^2 - 16}{2x} \) by factorising first.

Answer: \frac{4(x^2 - 4)}{2x} = \frac{4(x - 2)(x + 2)}{2x} = \frac{2(x - 2)(x + 2)}{x}

Question 3: Factorise the quadratic expression \(x^2 + 5x + 6\).

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

Question 4: Calculate the simplified form of \(\frac{9x^2 - 25}{3x - 5}\) by factorising first.

Answer: \frac{(3x - 5)(3x + 5)}{3x - 5} = 3x + 5

Question 5: Factorise the expression \(x^2 - 4x + 4\).

Answer: (x - 2)^2

Question 6: Solve for x: \( \frac{x^2 - 9}{x - 3} = 0 \), after factorising.

Answer: x = 3 or x = -3

Question 7: A rectangular garden has its length expressed as \(x^2 + 3x\) and width as \(x + 1\). Factorise the expressions to find the possible values of x if the area is to be maximised.

Answer: Factorise: length = x(x + 3), width = x + 1

Question 8: Factorise the expression \( 2x^2 + 8x \) completely.

Answer: 2x(x + 4)

Question 9: Construct a quadratic expression that factors to \( (x - 1)(x + 4) \).

Answer: x^2 + 3x - 4

Question 10: Identify and correct the mistake: Factorising the expression \(x^2 + 5x + 4\) was done as \(x^2 + 4x + x + 4\) leading to \(x(x + 4) + 1(x + 4)\).

Answer: Incorrect because the middle term is 5x, not 4x + x; correct factorisation is (x + 1)(x + 4).

Question 11: Challenge: Factorise \(3x^2 - 12x + 12\) and explain the steps involved.

Answer: Factor out 3: 3(x^2 - 4x + 4) = 3(x - 2)^2