Question 735007
You are trying to find (m+c)(m+c) so that the constant term is 1/4 and this must be {{{c^2=1/4}}}.  This would mean, {{{c=1/2}}}.  The question next is, will c=1/2 give you the m term that you need?  


You see, you want this:  {{{(m+c)(m+c)=m^2+mc +mc+c^2}}}
and {{{m^2+mc+mc+c^2=m^2+1*m+1/4}}}.


You must have both, {{{2mc=m}}} AND {{{c^2=1/4}.  How will this work?
The 2mc means that {{{2c=1}}}.  The c^2 as from before means {{{c=1/2}}}.  Using the 2c=1 result, solve for c and find {{{c=1/2}}}.  


There it is.  {{{c=1/2}}}.  
{{{m^2+m+1/4=(m+1/2)^2}}}.