Simplify the following. i) \( \frac{x+5+\frac{4}{x}}{\frac{1-7}{x}+\frac{12}{x^{2}}} \) ii) \( \frac{x+5+\frac{4}{x}}{1-\frac{7}{x}+\frac{12}{x^{2}}} \)

To simplify the expressions, let's tackle them one by one. i) For the expression \( \frac{x+5+\frac{4}{x}}{\frac{1-7}{x}+\frac{12}{x^{2}}} \): First, simplify the denominator: \[ \frac{1-7}{x} + \frac{12}{x^2} = \frac{-6}{x} + \frac{12}{x^2} = \frac{-6x + 12}{x^2} = \frac{12 - 6x}{x^2} \] Now, rewrite the entire expression: \[ \frac{x + 5 + \frac{4}{x}}{\frac{12 - 6x}{x^2}} = (x + 5 + \frac{4}{x}) \cdot \frac{x^2}{12 - 6x} \] Next, simplify the numerator: \[ x + 5 + \frac{4}{x} = \frac{x^2 + 5x + 4}{x} \] Combining this gives: \[ \frac{x^2 + 5x + 4}{x} \cdot \frac{x^2}{12 - 6x} = \frac{(x^2 + 5x + 4)x}{12 - 6x} \] This results in: \[ \frac{x^3 + 5x^2 + 4x}{12 - 6x} \] ii) For the expression \( \frac{x+5+\frac{4}{x}}{1-\frac{7}{x}+\frac{12}{x^{2}}} \): First, simplify the denominator: \[ 1 - \frac{7}{x} + \frac{12}{x^2} = \frac{x^2 - 7x + 12}{x^2} \] Now rewrite the expression: \[ \frac{x + 5 + \frac{4}{x}}{\frac{x^2 - 7x + 12}{x^2}} = (x + 5 + \frac{4}{x}) \cdot \frac{x^2}{x^2 - 7x + 12} \] We already know the numerator: \[ x + 5 + \frac{4}{x} = \frac{x^2 + 5x + 4}{x} \] Thus, we have: \[ \frac{x^2 + 5x + 4}{x} \cdot \frac{x^2}{x^2 - 7x + 12} = \frac{(x^2 + 5x + 4)x}{x^2 - 7x + 12} \] So, the simplified expression is: \[ \frac{x^3 + 5x^2 + 4x}{x^2 - 7x + 12} \] In summary: i) \( \frac{x^3 + 5x^2 + 4x}{12 - 6x} \) ii) \( \frac{x^3 + 5x^2 + 4x}{x^2 - 7x + 12} \)