To find the greatest common factor (GCF) of the expression \( 4x^2y^5 + 10x^3y^4 \), we need to determine the GCF of the coefficients and the GCF of the variables.
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Coefficients: The coefficients are 4 and 10. The GCF of 4 and 10 is 2.
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For \(x\): The terms contain \(x^2\) and \(x^3\). The GCF of \(x^2\) and \(x^3\) is \(x^2\).
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For \(y\): The terms contain \(y^5\) and \(y^4\). The GCF of \(y^5\) and \(y^4\) is \(y^4\).
Now, we combine these factors to find the GCF:
\[ \text{GCF} = 2 \cdot x^2 \cdot y^4 = 2x^2y^4 \]
So, the greatest common factor of the expression \( 4x^2y^5 + 10x^3y^4 \) is \( 2x^2y^4 \).
Thus, the correct response from the options given is:
2x^2y^4.