Step 1: Use the binomial expansion formula for $(a+b)^3$. The formula is $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$. In this case, $a=1$ and $b=\frac{2}{x}$. Step 2: Substitute the values of $a$ and $b$ into the formula. $$ \left(1 + \frac{2}{x}\right)^3 = (1)^3 + 3(1)^2\left(\frac{2}{x}\right) + 3(1)\left(\frac{2}{x}\right)^2 + \left(\frac{2}{x}\right)^3 $$ Step 3: Simplify each term. $$ = 1 + 3(1)\left(\frac{2}{x}\right) + 3(1)\left(\frac{4}{x^2}\right) + \left(\frac{8}{x^3}\right) $$ $$ = 1 + \frac{6}{x} + \frac{12}{x^2} + \frac{8}{x^3} $$ The expanded form is: $$ \boxed{1 + \frac{6}{x} + \frac{12}{x^2} + \frac{8}{x^3}} $$