Factorise using the difference between two squares: (x+2)^2-4 (x-3)^2-36 100-(x-11)^2 81-(x-8)^2 (x+7)^2-1 f 9-(x+4)^2 Answer the Opening Problem on page 208. 7 Factorise using the difference between two squares: (x+3)^2-(x-1)^2 (2x-1)^2-(x+1)^2 (1-3x)^2-(2x+1)^2

Let's factorize each expression using the difference of squares formula, which states that \(a^2 - b^2 = (a + b)(a - b)\).<br /><br />1. \((x+2)^{2} - 4\)<br /><br /> Here, \(a = x + 2\) and \(b = 2\).<br /><br /> \[<br /> (x+2)^{2} - 4 = (x+2)^{2} - 2^{2} = (x+2 + 2)(x+2 - 2) = (x+4)(x)<br /> \]<br /><br />2. \((x-3)^{2} - 36\)<br /><br /> Here, \(a = x - 3\) and \(b = 6\).<br /><br /> \[<br /> (x-3)^{2} - 36 = (x-3)^{2} - 6^{2} = (x-3 + 6)(x-3 - 6) = (x+3)(x-9)<br /> \]<br /><br />3. \(100 - (x-11)^{2}\)<br /><br /> Here, \(a = 10\) and \(b = x - 11\).<br /><br /> \[<br /> 100 - (x-11)^{2} = 10^{2} - (x-11)^{2} = (10 + x-11)(10 - x+11) = (x-1)(19-x)<br /> \]<br /><br />4. \(81 - (x-8)^{2}\)<br /><br /> Here, \(a = 9\) and \(b = x - 8\).<br /><br /> \[<br /> 81 - (x-8)^{2} = 9^{2} - (x-8)^{2} = (9 + x-8)(9 - x+8) = (x-1)(17-x)<br /> \]<br /><br />5. \((x+7)^{2} - 1\)<br /><br /> Here, \(a = x + 7\) and \(b = 1\).<br /><br /> \[<br /> (x+7)^{2} - 1 = (x+7)^{2} - 1^{2} = (x+7 + 1)(x+7 - 1) = (x+8)(x+6)<br /> \]<br /><br />6. \(9 - (x+4)^{2}\)<br /><br /> Here, \(a = 3\) and \(b = x + 4\).<br /><br /> \[<br /> 9 - (x+4)^{2} = 3^{2} - (x+4)^{2} = (3 + x+4)(3 - x-4) = (x+7)(-x-1)<br /> \]<br /><br />7. \((x+3)^{2} - (x-1)^{2}\)<br /><br /> Here, \(a = x + 3\) and \(b = x - 1\).<br /><br /> \[<br /> (x+3)^{2} - (x-1)^{2} = (x+3 + x-1)(x+3 - x+1) = (2x+2)(2) = 2(x+1)(x+2)<br /> \]<br /><br />8. \(({2} - (x+1)^{2}\)<br /><br /> Here, \(a = 2x - 1\) and \(b = x + 1\).<br /><br /> \[<br /> (2x-1)^{2} - (x+1)^{2} = (2x-1 + x+1)(2x-1 - x-1) = (3x)(x-2) = 3x(x-2)<br /> \]<br /><br />9. \((1-3x)^{2} - (2x+1)^{2}\)<br /><br /> Here, \(a = 1 - 3x\) and \(b = 2x + 1\).<br /><br /> \[<br /> (1-3x)^{2} - (2x+1)^{2} = (1-3x + 2x+1)(1-3x - 2x-1) = (-x+2)(-5x) = -5x(-x+2) = 5x(x-2)<br /> \]<br /><br />So, the factorized forms are:<br /><br />1. \((x+4)(x)\)<br />2. \((x+3)(x-9)\)<br />3. \((x-1)(19-x)\)<br />4. \((x-1)(17-x)\)<br />5. \((x+8)(x+6)\)<br />6. \((x+7)(-x-1)\)<br />7. \(2(x+1)(x+2)\)<br />8. \(3x(x-2