To factor the expression \( 4x^6y^4 - 20x^4y^3 + x^2y^2 \) by factoring out the greatest common factor (GCF), we first need to identify the GCF of the coefficients and the variable parts in each term.
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Coefficients: The coefficients are \( 4, -20, \) and \( 1 \). The GCF of these coefficients is \( 1 \).
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Variable parts:
- For \( x \), the powers are \( 6, 4, \) and \( 2 \). The smallest power is \( 2 \), so \( x^2 \) is part of the GCF.
- For \( y \), the powers are \( 4, 3, \) and \( 2 \). The smallest power is \( 2 \), so \( y^2 \) is part of the GCF.
So, the GCF of the entire expression is \( x^2y^2 \).
Now, we can factor \( x^2y^2 \) out of the expression:
\[ 4x^6y^4 - 20x^4y^3 + x^2y^2 = x^2y^2(4x^4y^2 - 20x^2y + 1) \]
Thus, the expression in factored form is:
\[ \boxed{x^2y^2(4x^4y^2 - 20x^2y + 1)} \]