To simplify or analyze the given expression:

$(2 - y)(y^2 + 1) + y^3 + 3y$

Step-by-step solution:

1. Expand $(2 - y)(y^2 + 1)$: Using the distributive property: $(2 - y)(y^2 + 1) = 2(y^2 + 1) - y(y^2 + 1)$ $= 2y^2 + 2 - y^3 - y$

   The expanded form of $(2 - y)(y^2 + 1)$ is: $-y^3 + 2y^2 - y + 2$

2. Add $y^3 + 3y$ to the expanded expression: Combine like terms: $-y^3 + 2y^2 - y + 2 + y^3 + 3y$ Group terms: $(-y^3 + y^3) + 2y^2 + (-y + 3y) + 2$ Simplify: $2y^2 + 2y + 2$

Final simplified expression:

$2y^2 + 2y + 2$

Would you like a deeper explanation of any step or more practice on similar problems? Here are five follow-up questions:

1. What happens when you factorize $2y^2 + 2y + 2$?
2. Can you evaluate this expression for a specific value of $y$, such as $y = 1$?
3. How would the expansion change if the initial expression had different coefficients?
4. What is the graphical representation of $2y^2 + 2y + 2$?
5. How does this expression relate to quadratic equations?

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