To factor the polynomial \(50m^3 - 25m^2 + 44m - 22\), we can start by grouping the terms.
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Grouping the polynomial: \[ (50m^3 - 25m^2) + (44m - 22) \]
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Factoring out the common factors from each group:
- From the first group, \(50m^3 - 25m^2\), we can factor out \(25m^2\): \[ 25m^2(2m - 1) \]
- From the second group, \(44m - 22\), we can factor out \(22\): \[ 22(2m - 1) \]
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Now we rewrite our original expression: \[ 25m^2(2m - 1) + 22(2m - 1) \]
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Notice that \( (2m - 1) \) is a common factor: \[ (2m - 1)(25m^2 + 22) \]
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Now we have factored the expression into: \[ (2m - 1)(25m^2 + 22) \]
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Checking for further factorization:
- \(25m^2 + 22\) does not factor further over the reals (since its roots involve complex numbers).
Thus, the completely factored form of the polynomial \(50m^3 - 25m^2 + 44m - 22\) is: \[ \boxed{(2m - 1)(25m^2 + 22)} \]