x^{2} - (m+1)x + m + 4 = 0

If the equation has ral roots, then the discriminant must be positive:

Let us verify:

d = b^{2} - 4ac > 0

==> (m+1)^{2} - 4*1* (m+4) > 0

==> m^{2} + 2m + 1 - 4m - 16 > 0

==> m^{2} -2m - 15 > 0

==> (m-5)(m+3) >0

==> m-5 > 0 and (m+3)>0 OR (m-5)< 0 and m+3 < 0

==> m > 5 and m > -3 OR m<5 and m<-3