Given #p(x) = 4 x^4 + 12 x^3 + x^2 - 12 x# we can rewrite as
#p(x) = 4(x+3/4)^4-25/2(x+3/4)^2+369/64# so #p(x)# has a symmetry axis at #x = -3/4#
Now considering #p(x) = a(x-x_0)^4+b(x-x_0)^2+c# we have
#(dp)/(dx) = 4a(x-x_0)^3+2b(x-x_0)# so the stationary points are located at
# 4a(x-x_0)^3+2b(x-x_0)=0->{(x-x_0=0),(4a(x-x_0)^2+2b=0):}#
so the stationary points are
#x_0 - sqrt(-b/(2a)), x_0, x_0 + sqrt(-b/(2a))#
By symmetry we have a local maximum at #x=-3/4# which is #369/64# and two symmetrical minima regarding #x=-3/4# at #-2# and #1/2# with value #p(-2)=p(-1/2) = -4#
then we can conclude
#{(alpha < -369/64->p(x)+alpha = 0 " have two real roots"),(-369/64 < alpha < 4->p(x)+alpha=0 " have four real roots"),(alpha gt 4->p(x)+alpha = 0 " does not have real roots"):}#
