What will be the value of `(2x - 3y)^3 + 18xy (2x - 3y)`?

Options

A. `8x^3 + 27y^3`
B. `8x^3 - 27y^3`
C. `8x^3 + 27 y^3 - 18 x^2y`
D. `8x^3 + 27y^3 - 18 xy^2`

Answer

To solve the expression \((2x - 3y)^3 + 18xy(2x - 3y)\), we can follow these steps:

### Step 1: Expand \((2x - 3y)^3\)
Using the formula for the cube of a binomial, \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\), we can expand \((2x - 3y)^3\).

Let \(a = 2x\) and \(b = 3y\):
\[ (2x - 3y)^3 = (2x)^3 - 3(2x)^2(3y) + 3(2x)(3y)^2 - (3y)^3 \]
Calculating each term:
- \((2x)^3 = 8x^3\)
- \(-3(2x)^2(3y) = -3 \cdot 4x^2 \cdot 3y = -36x^2y\)
- \(3(2x)(3y)^2 = 3 \cdot 2x \cdot 9y^2 = 54xy^2\)
- \(-(3y)^3 = -27y^3\)

Putting it all together:
\[ (2x - 3y)^3 = 8x^3 - 36x^2y + 54xy^2 - 27y^3 \]

### Step 2: Expand \(18xy(2x - 3y)\)
Now we expand \(18xy(2x - 3y)\):
\[ 18xy(2x - 3y) = 18xy \cdot 2x - 18xy \cdot 3y = 36x^2y - 54xy^2 \]

### Step 3: Combine the two results
Now we combine the results from Step 1 and Step 2:
\[ (2x - 3y)^3 + 18xy(2x - 3y) = (8x^3 - 36x^2y + 54xy^2 - 27y^3) + (36x^2y - 54xy^2) \]

### Step 4: Simplify the expression
Now we simplify by combining like terms:
- The \(x^3\) term: \(8x^3\)
- The \(x^2y\) terms: \(-36x^2y + 36x^2y = 0\)
- The \(xy^2\) terms: \(54xy^2 - 54xy^2 = 0\)
- The \(y^3\) term: \(-27y^3\)

Thus, we have:
\[ 8x^3 - 27y^3 \]

### Final Result
The final value of the expression \((2x - 3y)^3 + 18xy(2x - 3y)\) is:
\[ \boxed{8x^3 - 27y^3} \]