Factorisation

2x(2x + 3y)(2x - 3y)

First take out the common factor 2x: 8x^3 - 18xy^2 = 2x(4x^2 - 9y^2). Then use difference of squares: 4x^2 - 9y^2 = (2x + 3y)(2x - 3y). So the complete factorisation is 2x(2x + 3y)(2x - 3y).

3(x + 6)(x - 2)

First take out the common factor 3: 3x^2 + 12x - 36 = 3(x^2 + 4x - 12). Now split 4x into 6x - 2x, since 6 + (-2) = 4 and 6 × (-2) = -12. So x^2 + 4x - 12 = x^2 + 6x - 2x - 12 = x(x + 6) - 2(x + 6) = (x + 6)(x - 2). Therefore the answer is 3(x + 6)(x - 2).

(1 + 2a - 4b)(1 - 2a + 4b)

Write 4(a - 2b)^2 as [2(a - 2b)]^2 = (2a - 4b)^2. Then 1 - (2a - 4b)^2 is a difference of squares. So it factorises to (1 + 2a - 4b)(1 - 2a + 4b).

(x - 5)(x - 4)

Find two numbers whose product is 20 and sum is -9. They are -5 and -4. So x^2 - 9x + 20 = x^2 - 5x - 4x + 20 = x(x - 5) - 4(x - 5) = (x - 5)(x - 4).