Chứng minh

a. $\frac{5}{6 x^{2} y} + \frac{7}{12 x y^{2}} + \frac{11}{18 x y}$ $= \frac{30 y}{36 x^{2} y^{2}} + \frac{21 x}{36 x^{2} y^{2}} + \frac{22 x y}{36 x^{2} y^{2}} = \frac{30 y + 21 x + 22 x y}{36 x^{2} y^{2}}$

b. $\frac{4 x + 2}{15 x^{3} y} + \frac{5 y - 3}{9 x^{2} y} + \frac{x + 1}{5 x y^{3}}$ $\begin{matrix} = \frac{3 y^{2} (4 x + 2)}{45 x^{3} y^{3}} + \frac{5 x y^{2} (5 y - 3)}{45 x^{3} y^{3}} + \frac{9 x^{2} (x + 1)}{45 x^{3} y^{3}} = \frac{12 x y^{2} + 6 y^{2} + 25 x y^{3} - 15 x y^{2} + 9 x^{3} + 9 x^{2}}{45 x^{3} y^{3}} = \frac{6 y^{2} + 25 x y^{3} - 3 x y^{2} + 9 x^{3} + 9 x^{2}}{45 x^{3} y^{3}} \end{matrix}$

c. $\frac{3}{2 x} + \frac{3 x - 3}{2 x - 1} + \frac{2 x^{2} + 1}{4 x^{2} - 2 x}$ $= \frac{3}{2 x} + \frac{3 x - 3}{2 x - 1} + \frac{2 x^{2} + 1}{2 x (2 x - 1)}$

$\begin{matrix} = \frac{3 (2 x - 1)}{2 x (2 x - 1)} + \frac{2 x (3 x - 3)}{2 x (2 x - 1)} + \frac{2 x^{2} + 1}{2 x (2 x - 1)} = \frac{6 x - 3 + 6 x^{2} - 6 x + 2 x^{2} + 1}{2 x (2 x - 1)} = \frac{8 x^{2} - 2}{2 x (2 x - 1)} = \frac{2 (4 x^{2} - 1)}{2 x (2 x - 1)} = \frac{(2 x + 1) (2 x - 1)}{x (2 x - 1)} = \frac{2 x + 1}{x} \end{matrix}$

d. $\frac{x^{3} + 2 x}{x^{3} + 1} + \frac{2 x}{x^{2} - x + 1} + \frac{1}{x + 1}$ $= \frac{x^{3} + 2 x}{(x + 1) (x^{2} - x + 1)} + \frac{2 x}{x^{2} - x + 1} + \frac{1}{x + 1}$

$\begin{matrix} = \frac{x^{3} + 2 x}{(x + 1) (x^{2} - x + 1)} + \frac{2 x (x + 1)}{(x + 1) (x^{2} - x + 1)} + \frac{x^{2} - x + 1}{(x + 1) (x^{2} - x + 1)} = \frac{x^{3} + 2 x + 2 x^{2} + 2 x + x^{2} - x + 1}{(x + 1) (x^{2} - x + 1)} = \frac{x^{3} + 3 x^{2} + 3 x + 1}{(x + 1) (x^{2} - x + 1)} = \frac{{(x + 1)}^{3}}{(x + 1) (x^{2} - x + 1)} = \frac{{(x + 1)}^{2}}{x^{2} - x + 1} \end{matrix}$