To factor the expression \( 45x - 18y + 36 \) completely, we start by looking for the greatest common factor (GCF) of the terms.
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The coefficients of the terms are \( 45 \), \( -18 \), and \( 36 \). The GCF of these coefficients can be determined:
- The factors of \( 45 \) are \( 1, 3, 5, 9, 15, 45 \).
- The factors of \( -18 \) are \( -1, -2, -3, -6, -9, -18 \) (but we need the positive ones for GCF).
- The factors of \( 36 \) are \( 1, 2, 3, 4, 6, 9, 12, 18, 36 \).
The GCF of \( 45, -18, 36 \) is \( 9 \).
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Now, we can factor out \( 9 \) from the expression: \[ 45x - 18y + 36 = 9(5x - 2y + 4) \]
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Now we focus on the expression inside the parentheses, \( 5x - 2y + 4 \). We can check if this can be factored further:
- There are no common factors among \( 5x \), \( -2y \), and \( 4 \).
- We can try to factor it as a quadratic in terms of \( x \) or \( y \), but it doesn’t have any recognizable patterns (like being a perfect square or a difference of squares).
- Thus, it cannot be factored further.
Finally, the completely factored form of the expression is: \[ \boxed{9(5x - 2y + 4)} \]
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