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To multiply \((x^2 + 3xy - y)(3x + y)\) using polynomial long multiplication, we distribute each term in the first polynomial to each term in the second polynomial.

1. **Set up the expression**:

   \[
   (x^2 + 3xy - y)(3x + y)
   \]

2. **Distribute each term** in \((x^2 + 3xy - y)\) to each term in \((3x + y)\):

   - Multiply \(x^2\) by \(3x\) and \(y\):
     \[
     x^2 \cdot 3x = 3x^3
     \]
     \[
     x^2 \cdot y = x^2y
     \]

   - Multiply \(3xy\) by \(3x\) and \(y\):
     \[
     3xy \cdot 3x = 9x^2y
     \]
     \[
     3xy \cdot y = 3xy^2
     \]

   - Multiply \(-y\) by \(3x\) and \(y\):
     \[
     -y \cdot 3x = -3xy
     \]
     \[
     -y \cdot y = -y^2
     \]

3. **Combine all the terms**:

   \[
   3x^3 + x^2y + 9x^2y + 3xy^2 - 3xy - y^2
   \]

4. **Combine like terms** (if any):

   The like terms here are \(x^2y\) and \(9x^2y\):

   \[
   3x^3 + (x^2y + 9x^2y) + 3xy^2 - 3xy - y^2
   \]

   Simplify further:

   \[
   3x^3 + 10x^2y + 3xy^2 - 3xy - y^2
   \]

5. **Final Answer**:

   \[
   3x^3 + 10x^2y + 3xy^2 - 3xy - y^2
   \]

Would you like more details or have any questions on any of the steps?

Here are 5 related questions to extend your learning:
1. How do you multiply polynomials using the distributive property?
2. What are the steps for combining like terms in polynomial expressions?
3. Can polynomial long multiplication be used to verify the result of shorter multiplication methods, like FOIL?
4. How does polynomial multiplication apply to real-world contexts, such as calculating areas?
5. What are other methods to simplify complex polynomial expressions?

**Tip:** When multiplying polynomials, keep terms organized by writing them down clearly under each other to avoid mistakes in combining similar terms.

Multiply out (x^2 + 3xy - y)(3x + y) using polynomial long multiplication.

Learn how to expand the polynomial expression (x^2 + 3xy - y)(3x + y) using polynomial long multiplication. This problem involves key algebraic concepts, including distributing terms and combining like terms. Suitable for high school students in Grades 10-12.

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