$\dfrac{\left(\dfrac{\color{#026fc1}{x+1}}{\color{#026fc1}{x}}\right)}{\left(\dfrac{\color{#c1026f}{x-1}}{\color{#c1026f}{2x+1}}\right)} $
\t ${}=\dfrac{\color{#026fc1}{x+1}}{\color{#026fc1}{x}} \times \dfrac{\color{#c1026f}{2x+1}}{\color{#c1026f}{x-1}}$ \t
\\ \t ${}= \dfrac{\color{#026fc1}{(x+1)}\color{#c1026f}{(2x+1)}}{\color{#026fc1}{x}\color{#c1026f}{(x-1)}}$ \t
\\ \t ${}= \dfrac{2x^2+3x+1}{x^2-x}$
\\ \t
\\
$\dfrac{1}{\left(\dfrac{\color{#c1026f}{2x-1}}{\color{#c1026f}{x^3}}\right)}$
\t ${}= 1 \times \dfrac{\color{#c1026f}{x^3}}{\color{#c1026f}{2x-1}}$ \t
\\ \t ${}= \dfrac{x^3}{2x-1}$
\\
\\
$\dfrac{\left(\dfrac{\color{#026fc1}{4x}}{\color{#026fc1}{x-1}}\right)}{\color{#c1026f}{x^2+1}} $
\t ${}= \dfrac{\color{#026fc1}{4x}}{\color{#026fc1}{x-1}} \times \dfrac{\color{#c1026f}{1}}{\color{#c1026f}{x^2+1}} $ \t
\\ \t ${}= \dfrac{\color{#026fc1}{4x}\color{#c1026f}{(1)}}{\color{#026fc1}{(x-1)}\color{#c1026f}{(x^2+1)}}$ \t
\\ \t ${}= \dfrac{4x}{x^3-x^2+x+1}$ \t
\\ \t
\\ $\dfrac{\color{#c1026f}{x^2+1}}{\left(\dfrac{\color{#026fc1}{4x}}{\color{#026fc1}{x-1}}\right)} $ \t ${}= \color{#c1026f}{(x^2+1)} \times \dfrac{\color{#026fc1}{x-1}}{\color{#026fc1}{4x}}$ \t
\\ \t ${}= \dfrac{\color{#026fc1}{(x^2+1)}\color{#c1026f}{(x-1)}}{\color{#026fc1}{(1)}\color{#c1026f}{(4x)}}$ \t
\\ \t ${}= \dfrac{x^3-x^2+x+1}{4x}$ \t