To find the greatest common factor (GCF) of the expression $x^5y^4 + x^3y^2 + x^2y^3 + y$, we look for the common factors among all terms.

Given expression:

$x^5y^4 + x^3y^2 + x^2y^3 + y$

1. First term: $x^5y^4$
2. Second term: $x^3y^2$
3. Third term: $x^2y^3$
4. Fourth term: $y$

Now, we identify the common factors between the terms.

• The common factor for all terms in terms of $x$:

  • The first term has $x^5$, second has $x^3$, third has $x^2$, and the fourth has no $x$.
  • Therefore, the GCF for $x$ is 1, since the last term has no $x$.

• The common factor for all terms in terms of $y$:

  • The first term has $y^4$, second has $y^2$, third has $y^3$, and the fourth has $y$.
  • The GCF for $y$ is $y$, since $y$ is the smallest power present in all terms.

Thus, the greatest common factor (GCF) of the entire expression is:

$\boxed{y}$

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Would you like further details or have any questions? Here are five related questions:

1. How do you factor expressions with multiple variables?
2. What is the difference between GCF and LCM in algebra?
3. Can you always find a GCF when factoring polynomials?
4. How would you simplify the original expression after factoring out the GCF?
5. How do you approach factoring in expressions with negative terms?

Tip: When factoring expressions, always look for the smallest power of each variable present across all terms.