$$\eqalign{
& 1)\,\,{x^2} + 15x + 56 \cr
& = {x^2} + 7x + 8x + 56 \cr
& = \left( {{x^2} + 7x} \right) + \left( {8x + 56} \right) \cr
& = x\left( {x + 7} \right) + 8\left( {x + 7} \right) \cr
& = \left( {x + 7} \right)\left( {x + 8} \right) \cr
& 2)\,\,{x^2} - 9x + 20 \cr
& = {x^2} - 4x - 5x + 20 \cr
& = \left( {{x^2} - 4x} \right) - \left( {5x - 20} \right) \cr
& = x\left( {x - 4} \right) - 5\left( {x - 4} \right) \cr
& = \left( {x - 4} \right)\left( {x - 5} \right) \cr
& 3)\,\,{x^2} - 6x - 16 \cr
& = {x^2} - 6x + 9 - 25 \cr
& = {\left( {x - 3} \right)^2} - {5^2} \cr
& = \left( {x - 3 - 5} \right)\left( {x - 3 + 5} \right) \cr
& = \left( {x - 8} \right)\left( {x + 2} \right) \cr
& 4)\,\,{x^2} + x + {1 \over 4} \cr
& = {x^2} + 2.x.{1 \over 2} + {\left( {{1 \over 2}} \right)^2} \cr
& = {\left( {x + {1 \over 2}} \right)^2} \cr
& 5)\,\,{x^2} - x - {1 \over 4} \cr
& = {x^2} - x + {1 \over 4} - {1 \over 2} \cr
& = \left( {{x^2} - 2.x.{1 \over 2} + {{\left( {{1 \over 2}} \right)}^2}} \right) - {\left( {{1 \over {\sqrt 2 }}} \right)^2} \cr
& = {\left( {x - {1 \over 2}} \right)^2} - {\left( {{1 \over {\sqrt 2 }}} \right)^2} \cr
& = \left( {x - {1 \over 2} - {1 \over {\sqrt 2 }}} \right)\left( {x - {1 \over 2} + {1 \over {\sqrt 2 }}} \right) \cr} $$
