Example 5 (1987 National College Entrance Examination Question) Suppose for all real numbers $x$, the inequality $x^{2} \cdot \log _{2} \frac{4(a+1)}{a}+2 x \log _{2} \frac{2 a}{a+1}+\log _{2} \frac{(a+1)^{2}}{4 a^{2}}$ $>0$ always holds. Find the range of values for $a$;
Let $\log _{2} \frac{(a+1)}{a}=t$,
then $x^{2} \cdot \log _{2} \frac{4(a+1)}{a}+2 x \log _{2} \frac{2 a}{a+1}+\log _{2} \frac{(a+1)^{2}}{4 a^{2}}>0$ always holds
$\Leftrightarrow(3+t) x^{2}-2 t x+2 t>0$ always holds
$\Leftrightarrow t>-\frac{3 x^{2}}{x^{2}-2 x+2}$
$\Leftrightarrow$ Let $f(x)=-\frac{3 x^{2}}{x^{2}-2 x+2}, t>f_{\text {max }}(x)$.
$\because$ When $x \in \mathbf{R}$, $f(x) \leq 0$, and when $x=0$, $f_{\text {max }}(x)=0, \therefore t>0$, i.e., $\log _{2} \frac{(a+1)}{a}>0, \therefore a$'s range of values is $a \in(0,1)$.