Given
#f(x) = x^2+(h-3)x+4 > 0#
Analyzing #(xpm2)^2= x^2pm4x+4 ge 0# and comparing we have
#h-3 = pm 4# giving #h = {(-1),(7):}#
and then for # h in (-1,7)# we will guarantee that for all #x -> f(x) > 0#
NOTE:
Considering the polynomial
#y =x^2+alpha x+4 > 0# which means that the polynomial does not have real roots. Also #x^2# as the main term, indicates that the polynomial graph must be included in the semi-plane #y > 0# then in
#x = 1/2(-alphapm sqrt(alpha^2-16))#
#alpha^2 < 16# guarantees no real roots but
#alpha^2 < 16 rArr {(alpha < 4rArr h-3 < 4 hArr h < 7),(alpha > -4 rArr h-3 > -4 rArr h > -1 ):}#
or
#-1 < h < 7#
