Always Non-negative

try to convert this eqn. to such a form which contains (m + n). to do this, we have to simply put x = -(1/2). this gives us m + n +(21/4) > or equal to 0. this implies m + n is greater than or equal to -(21/4)

Since the inequality must be valid for an arbitrary value of $x$ , the discriminant of the equation has to be negative: $\frac{\Delta}{4}=m^2-(n+5)<0$ . Our goal is to find the minimum of the function $f(m, n) = m + n$ under the constraint $n>m^2-5$

It's easy to note that the minimum must occur on the constraint itself, so we can simply minimize $g(m)=f(m, m^2-5) = m^2+x-5$ : $g'(m) = 2m+1=0$ implies $m=-\frac{1}{2}$ and $n=-\frac{19}{4}$ So the answer is $m+n= -\frac{21}{4}$