To find the least common denominator (LCD) of the fractions \(\frac{6}{(m-2)(m+5)}\) and \(\frac{-4}{(m-2)(m-7)}\), follow these steps:
1. Identify the denominators of each fraction:
- The denominators are \((m-2)(m+5)\) and \((m-2)(m-7)\).
2. Factorize each denominator:
- The first denominator is already factored as \((m-2)(m+5)\).
- The second denominator is already factored as \((m-2)(m-7)\).
3. Determine the unique factors from both denominators:
- From \((m-2)(m+5)\), the factors are \(m-2\) and \(m+5\).
- From \((m-2)(m-7)\), the factors are \(m-2\) and \(m-7\).
4. Combine all unique factors to form the least common denominator:
- The common factor in both denominators is \(m-2\).
- The unique factors are \((m+5)\) from the first denominator and \((m-7)\) from the second denominator.
- Therefore, the LCD should include each of these factors only once: \(m-2\), \(m+5\), and \(m-7\).
5. Multiply these factors together to get the LCD:
- The least common denominator (LCD) is \((m-2)(m+5)(m-7)\).
Therefore, the least common denominator is [tex]\(\boxed{(m-2)(m+5)(m-7)}\)[/tex].
