Let's compress the y-axis

A standard approach will be to find the derivative of the above expression. If a function is strictly increasing or strictly decreasing along the real line then the function takes all real values.

In this case the derivative comes out to be $\large{\frac{8x^2-4\lambda x+5\lambda -12}{(4x-\lambda)^2}}$

We notice that the numerator part is an concave up parabola and the denominator is always greater than zero in the provided domain.

Thus the derivative must be greater than zero.

$\large{8x^2-4\lambda x+5\lambda -12>0}$

This implies that it's discriminant should be less than zero.

$\large{16 \lambda^2-32(5\lambda -12)<0}$

$\large{(\lambda-4)(\lambda-6)<0}$

$\large{\boxed{\lambda \in (4,6)}}$