[tex]e) \: (5x + 1)(5x - 1) ={ (3 - 5x) }^{2} \\ {(5x)}^{2} - {1}^{2} = {3}^{2} - 2 \times 3 \times 5x + {(5x)}^{2} \\ 25 {x}^{2} - 1 = 9 - 30x + 25 {x}^{2} \: \: | - 25 {x}^{2} \\ - 1 = 9 - 30x \: \: | + 30x \\ 30x - 1 = 9 \: \: | + 1 \\ 30x = 10 \: \: | \div 30 \\ x = \frac{1}{3} [/tex]
[tex]f) \: {(x + 2)}^{2} = x(x + 6) \\ {x}^{2} + 2 \times x \times 2 + {2}^{2} = {x}^{2} + 6x \\ {x}^{2} + 4x + 4 = {x}^{2} + 6x \: \: | - {x}^{2} \\ 4x + 4 = 6x \: \: | - 6x \\4 - 2x = 0 \: \: | - 4 \\ - 2x = - 4 \: \: | \div ( - 2) \\ x = 2[/tex]
[tex]g) \: 4 {x}^{2} (3 - 2x) = {(1 - 2x)}^{3} \\ 12 {x}^{2} - 8 {x}^{3} = {1}^{3} - 3 \times {1}^{2} \times 2x + 3 \times 1 \times {(2x)}^{2} - {(2x)}^{3} \\ 12 {x}^{2} - 8 {x}^{3} = 1 - 6x + 12 {x}^{2} - 8 {x}^{3} \: \: | - (12 {x}^{2} - 8 {x}^{3} ) \\ 0 = 1 - 6x \: \: | + 6x \\ 6x = 1 \: \: | \div 6 \\ x = \frac{1}{6} [/tex]
[tex]h) \: {(x - 2)}^{3} = x {(x - 3)}^{2} \\ {x}^{3} - 3 \times {x}^{2} \times 2 + 3 \times x \times {2}^{2} - {2}^{3} = x( {x}^{2} - 2 \times x \times 3 + {3}^{2} ) \\ {x}^{3} - 6 {x}^{2} + 12x - 8 = x( {x}^{2} - 6x + 9) \\ {x}^{3} - 6 {x}^{2} + 12x - 8 = {x}^{3} - 6 {x}^{2} + 9x \: \: | - ( {x}^{3} - 6 {x}^{2} ) \\ 12x - 8 = 9x \: \: | - 9x \\ 3x - 8 = 0 \: \: | + 8 \\ 3x = 8 \: \: | \div 3 \\ x = 2\frac{2}{3} [/tex]