[tex]42) \\ \\ a) {x}^{2} + 5x + 6 = {x}^{2} + 2x + 3x + 6 = x( \frac{ {x}^{2} }{x} + \frac{2x}{x}) + 3( \frac{3x}{3} + \frac{2 \times 3}{3}) = x( {x}^{2 - 1} + 2) + 3(x + 2) = x(x + 2) + 3(x + 2) = (x + 2)( \frac{x(x + 2)}{x + 2} + \frac{3(x + 2)}{x + 2}) = (x + 2)(x + 3) [/tex]
[tex]b) {x}^{2} + 7x + 10 = {x}^{2} + 2x + 5x + 10 = x( \frac{ {x}^{2} }{x} + \frac{2x}{x}) + 5( \frac{5x}{5} + \frac{2 \times 5}{5}) = x( {x}^{2 - 1} + 2) + 5(x + 2) = x(x + 2) + 5(x + 2) = (x + 2)( \frac{x(x + 2)}{x + 2} + \frac{5(x + 2)}{x + 2}) = (x + 2)(x + 5) [/tex]
[tex]c) {x}^{2} + 8x + 15 = {x}^{2} + 3x + 5x + 15 = x( \frac{ {x}^{2} }{x} + \frac{3x}{x}) + 5( \frac{5x}{5} + \frac{3 \times 5}{5}) = x( {x}^{2 - 1} + 3) + 5(x + 3) = x(x + 3) + 5(x + 3) = (x + 3)( \frac{x(x + 3)}{x + 3} + \frac{5(x + 3)}{x + 3}) = (x + 3)(x + 5) [/tex]
[tex]d) {x}^{2} + 7x + 6 = {x}^{2} + x + 6x + 6 = x( \frac{ {x}^{2} }{x} + \frac{x}{x}) + 2 \times 3( \frac{2 \times 3x}{2 \times 3} + \frac{2 \times 3}{2 \times 3}) = x( {x}^{2 - 1} + 1) + 6(x + 1) = x(x + 1) + 6(x + 1) = x(x + 1) + (2 \times 3)(x + 1) = (x + 1)( \frac{x(x + 1)}{x + 1} + \frac{2 \times 3(x + 1)}{x + 1}) = (x + 1)(x + 2 \times 3) = (x + 1)(x + 6) [/tex]
[tex]e) {x}^{2} + 9x + 20 = {x}^{2} + 4x + 5x + 20 = x( \frac{ {x}^{2} }{x} + \frac{ {2}^{2} \times x }{x}) + 5( \frac{5x}{5} + \frac{ {2}^{2} \times 5 }{5}) = x( {x}^{2 - 1} + {2}^{2}) + 5(x + {2}^{2}) = x(x + 4) + 5(x + 4) = (x + 4)(\frac{x(x + 4)}{x + 4} + \frac{5(x + 4)}{x + 4}) = (x + 4)(x + 5) [/tex]
[tex]f) {x}^{2} + 12x + 27 = {x}^{2} + 3x + 9x + 27 = x( \frac{ {x}^{2} }{x} + \frac{3x}{x}) + {3}^{2}( \frac{ {3}^{2} \times x }{ {3}^{2} } + \frac{ {3}^{3} }{ {3}^{2} }) = x( {x}^{2 - 1} + 3) + 9(x + {3}^{3 - 2}) = x(x + 3) + 9(x + 3) = x(x + 3) + {3}^{2}(x + 3) = (x + 3)( \frac{x(x + 3)}{x + 3} + \frac{ {3}^{2}(x + 3) }{x + 3}) = (x + 3)(x + {3}^{2}) = (x + 3)(x + 9) [/tex]
[tex]g) {x}^{2} + 13x + 36 = {x}^{2} + 4x + 9x + 36 = x( \frac{ {x}^{2} }{x} + \frac{ {2}^{2} \times x }{x}) + {3}^{2}( \frac{ {3}^{2} \times x }{ {3}^{2} } + \frac{ {2}^{2} \times {3}^{2} }{ {3}^{2} }) = x( {x}^{2 - 1} + {2}^{2}) + 9(x + {2}^{2}) = x(x + 4) + 9(x + 4) = x(x + 4) + {3}^{2}(x + 4) = (x + 4)( \frac{x(x + 4)}{x + 4} + \frac{ {3}^{2}(x + 4) }{x + 4}) = (x + 4)(x + {3}^{2}) = (x + 4)(x + 9) [/tex]
[tex]h) {x}^{2} + 11x + 10 = {x}^{2} + x + 10x + 10 = x( \frac{ {x}^{2} }{x} + \frac{x}{x}) + 2 \times 5( \frac{2 \times 5x}{2 \times 5} + \frac{2 \times 5}{2 \times 5}) = x( {x}^{2 - 1} + 1) + 10(x + 1) = x(x + 1) + 10(x + 1) = x(x + 1) + (2 \times 5)(x + 1) = (x + 1)( \frac{x(x + 1)}{x + 1} + \frac{2 \times 5(x + 1)}{x + 1}) = (x + 1)(x + 2 \times 5) = (x + 1)(x + 10) [/tex]
[tex]i){x}^{2} + 15x + 56 = {x}^{2} + 7x + 8x + 56 = x( \frac{ {x}^{2} }{x} + \frac{7x}{x}) + {2}^{3}( \frac{ {2}^{3} \times x }{ {2}^{3} } + \frac{ {2}^{3} \times 7 }{ {2}^{3} }) = x( {x}^{2 - 1} + 7) + 8(x + 7) = x(x + 7) + 8(x + 7) = x(x + 7) + {2}^{3}(x + 7) = (x + 7)( \frac{x(x + 7)}{x + 7} + \frac{ {2}^{3}(x + 7) }{x + 7}) = (x + 7)(x + {2}^{3}) = (x + 7)(x + 8) [/tex]
