The factors of `(2x ^(2) - x - 6) ^(2) - ( 2x ^(2) - 9x + 10)^(2)` are:

Options

A. `8 (x -2) (2x -1) ^(2)`
B. `16 (x -2) ^(2) (2x -1)`
C. `16 (x - 2) ( 2x -1)`
D. `16 (x -2) ^(2) (2x -1) ^(2)`

Answer

To factor the expression \((2x^2 - x - 6)^2 - (2x^2 - 9x + 10)^2\), we can use the difference of squares formula, which states that \(a^2 - b^2 = (a + b)(a - b)\).

### Step-by-Step Solution:

1. **Identify \(a\) and \(b\)**:
Let \(a = 2x^2 - x - 6\) and \(b = 2x^2 - 9x + 10\).

2. **Apply the difference of squares formula**:
We can rewrite the expression as:
\[ (2x^2 - x - 6)^2 - (2x^2 - 9x + 10)^2 = (a + b)(a - b) \]

3. **Calculate \(a + b\)**:
\[ a + b = (2x^2 - x - 6) + (2x^2 - 9x + 10) \]
Combine like terms:
\[ = 2x^2 + 2x^2 - x - 9x - 6 + 10 = 4x^2 - 10x + 4 \]

4. **Calculate \(a - b\)**:
\[ a - b = (2x^2 - x - 6) - (2x^2 - 9x + 10) \]
Distributing the negative sign:
\[ = 2x^2 - x - 6 - 2x^2 + 9x - 10 = 8x - 16 \]

5. **Factor out the common terms**:
We can factor \(4\) from \(4x^2 - 10x + 4\):
\[ = 4(x^2 - \frac{10}{4}x + 1) = 4(x^2 - \frac{5}{2}x + 1) \]
And factor \(8\) from \(8x - 16\):
\[ = 8(x - 2) \]

6. **Combine the factors**:
Now we have:
\[ (4(x^2 - \frac{5}{2}x + 1))(8(x - 2)) \]
Multiplying the constants gives:
\[ 32(x^2 - \frac{5}{2}x + 1)(x - 2) \]

7. **Final factorization**:
The final factorization of the original expression is:
\[ 32(x^2 - \frac{5}{2}x + 1)(x - 2) \]