The sign of $-4\eta ^2\cosh\beta\cosh(\beta\eta)-4\eta\sinh\beta\sinh(\beta\eta)+2B^2\cosh(2\beta\eta)+2B^2+4$

Question

Can I figure out when the sign of this expression is positive and when it is negative?

$$-4 \eta ^2 \cosh (\beta ) \cosh (\beta \eta )-4 \eta \sinh (\beta ) \sinh (\beta \eta )+2 B^2 \cosh (2 \beta \eta )+2 B^2+4$$ where $\eta =\sqrt{B^2+1}$, and $\beta>0$.

Answer 1

Hint #1: $\eta$ (by definition) is positive for all values of $B$, so everything to the right of the hyperbolic functions will be positive.

Hint #2: Factor out $-4 \eta$ and take note of the signs of each of the hyperbolic equations.