Given $f\left(x\right) = \frac{4x^{2}+1}{x}$ Thus $f'\left(x\right) = 4-\frac{1}{x^{2}}$
$f\left(x\right)$ will be decreasing if $f'\left(x\right) < 0$
Thus $4-\frac{1}{x^{2}} < 0$
$\Rightarrow \frac{1}{x^{2}} > 4 \Rightarrow \frac{-1}{2} < x 1 < \frac{1}{2}$
Thus interval in which $f\left(x\right)$ is decreasing, is $\left(-\frac{1}{2}, \frac{1}{2}\right)$.