Divide: \( \frac{\frac{4}{x+5}-\frac{6}{x+3}}{-\frac{1}{x+5}} \)

To simplify the expression \[ \frac{\frac{4}{x+5}-\frac{6}{x+3}}{-\frac{1}{x+5}}, \] we start by simplifying the numerator, which is \[ \frac{4}{x+5} - \frac{6}{x+3}. \] To combine these fractions, we need a common denominator. The common denominator for \(x+5\) and \(x+3\) is \((x+5)(x+3)\). We rewrite each fraction with this common denominator: \[ \frac{4}{x+5} = \frac{4(x+3)}{(x+5)(x+3)} = \frac{4x + 12}{(x+5)(x+3)}, \] and \[ \frac{6}{x+3} = \frac{6(x+5)}{(x+5)(x+3)} = \frac{6x + 30}{(x+5)(x+3)}. \] Now we can combine the two fractions: \[ \frac{4}{x+5} - \frac{6}{x+3} = \frac{4x + 12 - (6x + 30)}{(x+5)(x+3)} = \frac{4x + 12 - 6x - 30}{(x+5)(x+3)} = \frac{-2x - 18}{(x+5)(x+3)}. \] Thus, the numerator simplifies to \[ \frac{-2(x + 9)}{(x+5)(x+3)}. \] Now we substitute this back into our original expression: \[ \frac{\frac{-2(x + 9)}{(x+5)(x+3)}}{-\frac{1}{x+5}}. \] Dividing by a fraction is equivalent to multiplying by its reciprocal: \[ = \frac{-2(x + 9)}{(x+5)(x+3)} \cdot \left(-\frac{x+5}{1}\right) = \frac{-2(x + 9)(-1)(x + 5)}{(x+5)(x+3)}. \] The negative signs cancel out, and we can simplify: \[ = \frac{2(x + 9)(x + 5)}{(x + 5)(x + 3)}. \] Now, we can cancel the \(x + 5\) terms in the numerator and denominator (assuming \(x \neq -5\)): \[ = \frac{2(x + 9)}{x + 3}. \] Thus, the final simplified expression is \[ \boxed{\frac{2(x + 9)}{x + 3}}. \]