Simplify using following identiy:
`(a +- b) (a^(2) ab + b^(2)) = a^(3) +- b^(3)`
`(2x + 3y) (4x^(2) - 6xy + 9y^(2))`

Answer

To simplify the expression \((2x + 3y)(4x^2 - 6xy + 9y^2)\) using the identity \((a \pm b)(a^2 \mp ab + b^2) = a^3 \pm b^3\), we can follow these steps:

### Step 1: Identify \(a\) and \(b\)
In our case, we can identify:
- \(a = 2x\)
- \(b = 3y\)

### Step 2: Check the format of the second term
We need to check if the second term \(4x^2 - 6xy + 9y^2\) matches the format \(a^2 - ab + b^2\):
- \(a^2 = (2x)^2 = 4x^2\)
- \(b^2 = (3y)^2 = 9y^2\)
- \(ab = (2x)(3y) = 6xy\)

Thus, \(4x^2 - 6xy + 9y^2\) matches the format \(a^2 - ab + b^2\).

### Step 3: Apply the identity
Since we have confirmed that the expression fits the identity, we can apply it:
\[ (2x + 3y)(4x^2 - 6xy + 9y^2) = (2x)^3 + (3y)^3 \]

### Step 4: Calculate \(a^3\) and \(b^3\)
Now, we calculate \(a^3\) and \(b^3\):
- \(a^3 = (2x)^3 = 8x^3\)
- \(b^3 = (3y)^3 = 27y^3\)

### Step 5: Combine the results
Putting it all together, we have:
\[ (2x + 3y)(4x^2 - 6xy + 9y^2) = 8x^3 + 27y^3 \]

### Final Answer
Thus, the simplified expression is:
\[ \boxed{8x^3 + 27y^3} \]

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