Hi

g(x)=-2x²+4x+6 = -2(x+1)(x-3)

It must be strictly positive because ln(a) is defined for a>0

The domain is therefore ]-1,3[

g(x)=-2x²+4x+6 is a parabola oriented towards negative y

Maximum is obtained for x=1 and g(1)=8

Range is therefore ]-oo,3ln(2)]