To factor out the greatest common factor (GCF) from the polynomial \(24m^7 - 4m^6 - 12m^5\), we first identify the GCF of the coefficients and the variable terms.
- Coefficients: The coefficients are 24, -4, and -12. The GCF of these coefficients is 4.
- Variable part: The terms have \(m^7\), \(m^6\), and \(m^5\). The smallest power of \(m\) is \(m^5\).
Combining both parts, the GCF of the entire polynomial is \(4m^5\).
Now we can factor out \(4m^5\):
\[ 24m^7 - 4m^6 - 12m^5 = 4m^5(6m^2 - m - 3) \]
Thus, the factored form of the polynomial is:
\[ \boxed{4m^5(6m^2 - m - 3)} \]
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