For some reason I need to find the values of two positive integers $k$ and $x$ such that:
$$ k x^3 (2 x + 1)^3>2 (k + 1) (x + 1)^2 (2 x^4 + 2 x^3 + 3 x^2 + 2 x + 1). $$
Inputting this in Wolfram Alpha gives the following strange result (among others):
$$ k>1, x>Root[\#1^6 (4 k - 4) - 12 \#1^5 + \#1^4 (-12 k - 18) + \#1^3 > (-19 k - 20) + \#1^2 (-16 k - 16) + \#1 (-8 k - 8) - 2 k - 2\&, 2] $$
What does this mean? What is the role of the symbols $\#$ and $\&$?
Using a command like
RegionPlot[ $k x^3 (2 x + 1)^3 - 2 (k + 1) (x + 1)^2 (2 x^4 + 2 x^3 + 3 x^2 + 2 x + 1) > 0$, {$k$, 0, 10}, {$x$, 0, 10}]
You will obtain an useful plot