You are asked to complete the square on $\simplify{{a}x^2+{b}x + {c}}$

First, divide the first two terms by 2 to get

\[\simplify{{a}x^2+{b}x + {c}} = \simplify[fractionNumbers,all, !collectNumbers]{{a}(x^2+{b/a}x) + {c}}\]

The exponent of x inside the bracket is $\var[fractionNumbers,all]{b/a}$, so we halve it to get $\var[fractionNumbers]{p}$.

Squaring this gives $\var[fractionNumbers]{p^2}$.

We can write 

\[\simplify{{a}x^2+{b}x + {c}} = \simplify[fractionNumbers,basic,unitFactor,unitDenominator,simplifyFractions]{{a}(x^2+{b/a}x + {p^2} - {p^2}) + {c}}\]

The first three terms inside the bracket are a perfect square, so we have

\[\simplify{x^2+{b}x + {c}} = \simplify[fractionNumbers,all,!collectNumbers]{{a}[(x+{p})^2 - {p^2}] + {c}}\]

Leading to

\[\simplify{x^2+{b}x+{c}} = \simplify[fractionNumbers,all,!collectNumbers]{{a}(x+{p})^2 - {a}*{p^2} + {c}}\]

Which finally gives

\[\simplify{x^2+{b}x+{c}} = \simplify[fractionNumbers,all]{(x+{p})^2 - {a*p^2} + {c}}\]