Factorise the following:

2x² + 13xy - 24y²

To factorise 2x² + 13xy - 24y², we want to find two real numbers whose sum is 13 and product is 2 × (-24) = -48. By trial, we see that 16 + (-3) = 13 and 16 × (-3) = -48.

∴ 2x² + 13xy - 24y² = 2x² + 16xy + (-3xy) - 24y²

= 2x² + 16xy - 3xy - 24y²

= 2x(x + 8y) - 3y(x + 8y)

= (x + 8y)(2x - 3y).

Hence, 2x² + 13xy - 24y² = (x + 8y)(2x - 3y).