I.
1
[tex] \frac{5}{x - 1} + \frac{3}{x - 1} =\red{\boxed{ \frac{8}{x - 1}}} [/tex]
2.
[tex] \frac{9 {x}^{2} }{3x + 1} - \frac{1}{3x + 1} \\ = \frac{9 {x }^{2} - 1 }{3x + 1} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \ = \frac{(3x + 1)(3x - 1)}{3x + 1} \\=\pink{\boxed{ 3x - 1}}[/tex]
3.
[tex] \frac{ {x}^{2} }{x - 1} + \frac{1}{1 - x} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \frac{ {x}^{2} (1 - x) + (x - 1)}{(x - 1)(1 - x)} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \frac{( {x}^{2} - 1)(1 - x)}{ {x }^{2} - 1 } \\ =\blue{\boxed{ 1 - x}} \: \: \: \: \: \: \: \: [/tex]
II.
1. [tex]\purple{\boxed{50 {x}^{5} {y}^{7}}} [/tex]
2.[tex]\green{\boxed{{(x - 8})^{2} (x + 8)}}[/tex]
III.
1. [tex] \frac{ {x}^{2} + 4x - 5}{ {x}^{2} - 2x - 3} - \frac{2}{x + 1} = \frac{2(x - 1)}{(x - 3)(x - 1)} =\orange{\boxed{ \frac{2}{x - 3}}} [/tex]
2.[tex] \frac{m + 1}{ {m}^{2} - 25 } + \frac{2m}{m - 5} = {\boxed{ \frac{2 {m}^{2} + 11m + 1}{ {m}^{2} - 25 }}} [/tex]
