To simplify the expression $(3x + y)(x^2 + 4y^2 + 2xy)$, we will apply the distributive property (also known as the FOIL method for binomials). Here are the steps:

$(3x + y)(x^2 + 4y^2 + 2xy)$

1. Multiply $3x$ by each term inside the second parentheses:

   • $3x \cdot x^2 = 3x^3$
   • $3x \cdot 4y^2 = 12xy^2$
   • $3x \cdot 2xy = 6x^2y$

2. Multiply $y$ by each term inside the second parentheses:

   • $y \cdot x^2 = x^2y$
   • $y \cdot 4y^2 = 4y^3$
   • $y \cdot 2xy = 2xy^2$

Now, combine all the terms:

$3x^3 + 12xy^2 + 6x^2y + x^2y + 4y^3 + 2xy^2$

Next, group and combine like terms:

$3x^3 + (12xy^2 + 2xy^2) + (6x^2y + x^2y) + 4y^3$

This simplifies to:

$3x^3 + 14xy^2 + 7x^2y + 4y^3$

Thus, the simplified expression is:

$3x^3 + 14xy^2 + 7x^2y + 4y^3$

Would you like a more detailed explanation or have any questions?

Here are 5 related questions to expand on this:

1. How do you apply the distributive property to binomials?
2. Can you show an example of simplifying a similar expression with different terms?
3. What is the difference between like and unlike terms in algebraic expressions?
4. How do you factor complex polynomials like the one above?
5. How would you apply this method to expressions with more variables?

Tip: Always look for like terms when combining algebraic expressions, as this can significantly simplify your calculations.