To find the perimeter of the polygon with side lengths \(3x + 7\), \(5x - 1\), \(2x\), \(4x\), and \(x - 1\), we will sum these expressions.
Let's add them together:
\[ (3x + 7) + (5x - 1) + (2x) + (4x) + (x - 1) \]
Now, we will combine like terms:
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Combine the \(x\) terms: \[ 3x + 5x + 2x + 4x + x = (3 + 5 + 2 + 4 + 1)x = 15x \]
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Combine the constant terms: \[ 7 - 1 - 1 = 7 - 2 = 5 \]
Putting it all together, the perimeter \(P\) is:
\[ P = 15x + 5 \]
Thus, the perimeter of the polygon is \( \boxed{15x + 5} \).