To solve the equation \( 4p + 25 = 6(p - 3) - 3(4 - 3p) \), we will first expand the right side of the equation.
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Start with the right side: \[ 6(p - 3) = 6p - 18 \] and \[ -3(4 - 3p) = -12 + 9p. \]
Now, combine these results: \[ 6(p - 3) - 3(4 - 3p) = 6p - 18 - 12 + 9p = 6p + 9p - 18 - 12 = 15p - 30. \]
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Now we rewrite the equation: \[ 4p + 25 = 15p - 30. \]
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Next, we will move all terms involving \( p \) to one side and constant terms to the other side. Subtract \( 4p \) from both sides: \[ 25 = 15p - 4p - 30, \] which simplifies to: \[ 25 = 11p - 30. \]
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Now, add 30 to both sides: \[ 25 + 30 = 11p, \] which simplifies to: \[ 55 = 11p. \]
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Finally, divide both sides by 11: \[ p = \frac{55}{11} = 5. \]
The solution to the equation is: \[ \boxed{5}. \]