Which expression is equivalent to this polynomial expression? (2x^2 – 3y^2)(4x^4 + 6x^2y^2 +9y^4)​

To find the expression equivalent to the given polynomial expression \((2x^2 - 3y^2)(4x^4 + 6x^2y^2 + 9y^4)\), we'll expand it using the distributive property.

### Step 1: Distribute each term in the first polynomial to every term in the second polynomial.

\[
(2x^2 - 3y^2)(4x^4 + 6x^2y^2 + 9y^4)
\]

This means we will distribute as follows:

\[
= 2x^2(4x^4 + 6x^2y^2 + 9y^4) - 3y^2(4x^4 + 6x^2y^2 + 9y^4)
\]

### Step 2: Multiply each term

1. **First distribution:**

\[
2x^2 \cdot 4x^4 = 8x^6
\]
\[
2x^2 \cdot 6x^2y^2 = 12x^4y^2
\]
\[
2x^2 \cdot 9y^4 = 18x^2y^4
\]

2. **Second distribution:**

\[
-3y^2 \cdot 4x^4 = -12x^4y^2
\]
\[
-3y^2 \cdot 6x^2y^2 = -18x^2y^4
\]
\[
-3y^2 \cdot 9y^4 = -27y^6
\]

### Step 3: Combine all the terms

Now, sum up all the terms obtained from the multiplication:

\[
8x^6 + 12x^4y^2 + 18x^2y^4 - 12x^4y^2 - 18x^2y^4 - 27y^6
\]

### Step 4: Simplify the expression

Combine like terms:

\[
8x^6 + (12x^4y^2 - 12x^4y^2) + (18x^2y^4 - 18x^2y^4) - 27y^6
\]

This simplifies to:

\[
8x^6 - 27y^6
\]

### Final Answer:
The expression equivalent to \((2x^2 - 3y^2)(4x^4 + 6x^2y^2 + 9y^4)\) is:

\[
8x^6 - 27y^6
\]

Would you like any additional details or have any further questions?

Here are five related questions you might find interesting:
1. How do you factor a polynomial like \(8x^6 - 27y^6\)?
2. What is the significance of the distributive property in polynomial multiplication?
3. How do you recognize and solve special products like the difference of squares?
4. What methods can be used to expand binomials raised to a power?
5. Can the polynomial \(8x^6 - 27y^6\) be factored further?

**Tip:** When expanding polynomial expressions, always group like terms together to simplify the expression more efficiently.

Explore the equivalent expression of (2x^2 - 3y^2)(4x^4 + 6x^2y^2 + 9y^4) through detailed step-by-step expansion using the distributive property. Understand how to simplify polynomial expressions and combine like terms effectively. Suitable for students in Grades 10-12.

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