Hind the least common denominator of \( \frac{x}{x-5} \) and \( \frac{4 x}{x-3} \), \( \square \)

To find the least common denominator (LCD) of the fractions \( \frac{x}{x-5} \) and \( \frac{4x}{x-3} \), we need to look at the denominators of both fractions, which are \( x - 5 \) and \( x - 3 \). The LCD is simply the product of these two distinct linear factors, which would be \( (x - 5)(x - 3) \). Thus, the least common denominator is \( (x - 5)(x - 3) \). However, always double-check any algebraic expressions for factoring or simplification opportunities before assuming you've got the final answer. In cases where there's a common factor among more than two fractions, ensure you're not missing a simpler LCD that incorporates any shared factors.