Factor the expression completely.x, to the power 4 , y, plus, y, to the power 5x 4 y+y 5

To factor the expression $x^4 y + y^5 x^4 y + y^5$ completely, we'll follow these steps:

1. Break Down the Problem

We need to identify common factors in the expression and factor them out. The expression can be organized as:

• $x^4 y + y^5 x^4 y + y^5$

2. Relevant Concepts

Identifying common factors is crucial. Here, we can observe that both $x^4 y$ and $y^5$ share some common terms.

3. Analysis and Detail

Looking at the expression:

• The first term is $x^4 y$
• The second term $y^5 x^4 y$ can be simplified and rewritten as $x^4 y^6$
• The third term is $y^5$

The expression becomes: $x^4 y + x^4 y^6 + y^5$

Next, rearranging helps us see the common factor: $x^4 y + y^5 (x^4 y + 1)$

4. Verify and Summarize

The step of factoring out $y^5$: $y^5 (x^4 y + 1)$ Finally, we look for further factors if possible.

Final Answer

The completely factored form of the expression is: $x^4 y + y^5 (x^4 y + 1) = y^5 (x^4 y + 1)$