(X+2y)(x square-3y)(4x+y) multiply the following 

To multiply the given expression \((x + 2y)(x^2 - 3y)(4x + y)\), we'll proceed step by step, first multiplying two of the binomials and then multiplying the result by the third.

### Step 1: Multiply \((x + 2y)\) and \((x^2 - 3y)\)

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

Distribute each term in \((x + 2y)\) over \((x^2 - 3y)\):

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

\[
= x^3 - 3xy + 2x^2y - 6y^2
\]

So the product is:
\[
x^3 - 3xy + 2x^2y - 6y^2
\]

### Step 2: Multiply the result by \((4x + y)\)

\[
(x^3 - 3xy + 2x^2y - 6y^2)(4x + y)
\]

Distribute each term in the polynomial \((x^3 - 3xy + 2x^2y - 6y^2)\) over \((4x + y)\):

\[
= (x^3 \cdot 4x) + (x^3 \cdot y) + (-3xy \cdot 4x) + (-3xy \cdot y) + (2x^2y \cdot 4x) + (2x^2y \cdot y) + (-6y^2 \cdot 4x) + (-6y^2 \cdot y)
\]

\[
= 4x^4 + x^3y - 12x^2y - 3xy^2 + 8x^3y + 2x^2y^2 - 24xy^2 - 6y^3
\]

### Step 3: Combine like terms

Now, combine like terms:

\[
4x^4 + (x^3y + 8x^3y) + (-12x^2y + 2x^2y^2) + (-3xy^2 - 24xy^2) - 6y^3
\]

\[
4x^4 + 9x^3y - 12x^2y + 2x^2y^2 - 27xy^2 - 6y^3
\]

### Final Answer:
\[
\boxed{4x^4 + 9x^3y - 12x^2y + 2x^2y^2 - 27xy^2 - 6y^3}
\]

Would you like to go over any part of this process in more detail?

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Here are some related questions:
1. How do you multiply polynomials with more than two terms?
2. What strategies can you use to simplify polynomial multiplication?
3. Can you explain how to group like terms effectively in a polynomial?
4. What is the importance of order in polynomial multiplication?
5. How do you multiply three binomials efficiently?

**Tip:** When multiplying several polynomials, it helps to work systematically from left to right, multiplying two polynomials at a time to avoid confusion.

Learn how to multiply the expression (x + 2y)(x^2 - 3y)(4x + y) with a detailed step-by-step solution. This problem covers polynomial multiplication involving multiple binomials, suitable for advanced high school students.

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