How to multiply $\sqrt{5x^2+2x+1}$ with $\frac{1}{x}$? I believe the answer says, $\sqrt{5+\frac{2}{x}+\frac{1}{x^2}}$, but I don't see how they obtained it. I'm quite rusty with radicals.

Just a slight variation to the approaches already posted.
Suppose 
$A = \sqrt{5x^2 + 2x + 1}$
$B = \frac{1}{x}$
If $AB = C$, then ${(AB)}^2 = C^2$ or $A^2B^2 = C^2$
$A^2$ = $5x^2 + 2x + 1$
$B^2$ = $\frac{1}{x^2}$
So $C^2 = 5 + \frac{2}{x} + \frac{1}{x^2}$
And to return to C: $$\sqrt{C^2} = \pm\sqrt{5 + \frac{2}{x} + \frac{1}{x^2}}$$
