<s>[INST] 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.
 [/INST] $f(x) = \sqrt{5 x^2 + 2x +1} $ is the positive function whose square $f^2$ is $f^2=5 x^2 + 2x +1$.
Hence the square $(\frac f x)^2$ is $(\frac f x)^2 = 5 +\frac 2 x + \frac 1 {x^2}$, which implies that 
$$\Big | \frac{\sqrt{ 5x^2 +2x+1}}{x}\Big|= \sqrt { 5 +\frac 2 x + \frac 1 {x^2}}.$$
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