Simplify `(2x-5y)^(3)-(2x+5y)^(3)`.

Options

A. `120x^(2)y-250y^(3)`
B. `120x^(2)y+250y^(3)`
C. `-120x^(2)y-250y^(3)`
D. can not determine its value

Explanation

To simplify the expression \((2x - 5y)^3 - (2x + 5y)^3\), we can use the identity for the difference of cubes:

\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) \]

Here, let:
- \(A = (2x - 5y)\)
- \(B = (2x + 5y)\)

### Step 1: Calculate \(A - B\)

\[ A - B = (2x - 5y) - (2x + 5y) = 2x - 5y - 2x - 5y = -10y \]

### Step 2: Calculate \(A^2\)

\[ A^2 = (2x - 5y)^2 = (2x)^2 - 2 \cdot (2x)(5y) + (5y)^2 = 4x^2 - 20xy + 25y^2 \]

### Step 3: Calculate \(B^2\)

\[ B^2 = (2x + 5y)^2 = (2x)^2 + 2 \cdot (2x)(5y) + (5y)^2 = 4x^2 + 20xy + 25y^2 \]

### Step 4: Calculate \(AB\)

\[ AB = (2x - 5y)(2x + 5y) = (2x)^2 - (5y)^2 = 4x^2 - 25y^2 \]

### Step 5: Combine \(A^2\), \(AB\), and \(B^2\)

Now we can combine these results:

\[ A^2 + AB + B^2 = (4x^2 - 20xy + 25y^2) + (4x^2 - 25y^2) + (4x^2 + 20xy + 25y^2) \]

Combine like terms:

\[ = 4x^2 + 4x^2 + 4x^2 + (-20xy + 20xy) + (25y^2 - 25y^2) \]
\[ = 12x^2 \]

### Step 6: Substitute back into the difference of cubes formula

Now substituting back into the difference of cubes formula:

\[ (2x - 5y)^3 - (2x + 5y)^3 = (A - B)(A^2 + AB + B^2) = (-10y)(12x^2) \]

### Step 7: Final result

Thus, the simplified expression is:

\[ = -120xy \]