Factariser \( (x-2)^{2}-9 x^{2} \)

To factor the expression \( (x-2)^{2}-9 x^{2} \), we start by recognizing it as a difference of squares. The first term, \( (x-2)^{2} \), and the second term, \( 9x^{2} \), can be seen as \( a^{2} - b^{2} \) where \( a = (x-2) \) and \( b = 3x \). Using the difference of squares formula \( a^{2} - b^{2} = (a-b)(a+b) \), we can apply it here: 1. \( a - b = (x-2) - 3x = -2x - 2 = -2(x + 1) \) 2. \( a + b = (x-2) + 3x = 4x - 2 = 2(2x - 1) \) Now, we can combine these to form the completely factored expression: \[ (x-2)^{2}-9 x^{2} = (a-b)(a+b) = (-2(x+1))(2(2x-1)) = -4(x+1)(2x-1) \] And there you have it! The factored form is \( -4(x+1)(2x-1) \). Happy factoring!