To factor the expression \( 8y^2 - 162 \), we can follow these steps:
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Factor out the greatest common factor (GCF): The GCF of \( 8y^2 \) and \( 162 \) is \( 2 \). \[ 8y^2 - 162 = 2(4y^2 - 81) \]
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Recognize the difference of squares: The expression \( 4y^2 - 81 \) is a difference of squares, which can be factored using the formula \( a^2 - b^2 = (a - b)(a + b) \). Here, \( a = 2y \) and \( b = 9 \) since \( 4y^2 = (2y)^2 \) and \( 81 = 9^2 \).
\[ 4y^2 - 81 = (2y - 9)(2y + 9) \]
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Combine it all together: Now we substitute back into our expression: \[ 8y^2 - 162 = 2(2y - 9)(2y + 9) \]
Thus, the completely factored form of the given expression is: \[ \boxed{2(2y - 9)(2y + 9)} \]