y = x^2 - mx + m + 3
= (x - m/2)^2 - m^2/4 + m + 3

頂点は(m/2 , - m^2/4 + m + 3)
だから
m/2 > 0 , - m^2/4 + m + 3 > 0
m > 0 , m^2 - 4m - 12 < 0
m > 0 , (m-6)(m+2) < 0
m > 0 , -2 < m < 6

0 < m < 6