To find the greatest common factor (GCF) of the expression \(4x^2 y^5 + 10x^3 y^4\), we will look at the coefficients and the variables separately.
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Coefficients:
- The coefficients are 4 and 10.
- The GCF of 4 and 10 is 2.
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Variables:
- For \(x^2\) and \(x^3\), the GCF is \(x^2\) (the lowest power of \(x\)).
- For \(y^5\) and \(y^4\), the GCF is \(y^4\) (the lowest power of \(y\)).
Now, we combine the GCFs of the coefficients and the variables:
\[ \text{GCF} = 2x^2y^4 \]
Thus, the greatest common factor of the expression \(4x^2 y^5 + 10x^3 y^4\) is:
\[ \boxed{2x^2y^4} \]