Factors of `(2x ^(2) - 3x - 2) ( 2x ^(2) - 3x) - 63` are :

Options

A. `(x-3) ( 2x + 3) ( x-1) ( x -7)`
B. ` (x +3) (2x + 3) ( 2x ^(2) - 1) (x - 7)`
C. `(x + 3) ( 2x + 3) ( 2x ^(2) - 2x + 7)`
D. `(x-3) (2x + 3) ( 2x ^(2) - 3x + 7)`

Answer

To factor the expression \( (2x^2 - 3x - 2)(2x^2 - 3x) - 63 \), we will follow these steps:

### Step 1: Set up the expression
We start with the expression:
\[ (2x^2 - 3x - 2)(2x^2 - 3x) - 63 \]

### Step 2: Substitute a variable
Let \( a = 2x^2 - 3x \). Then, we can rewrite the expression as:
\[ (a - 2)(a) - 63 \]

### Step 3: Expand the expression
Now, we expand the expression:
\[ a(a - 2) - 63 = a^2 - 2a - 63 \]

### Step 4: Factor the quadratic expression
Next, we need to factor \( a^2 - 2a - 63 \). We look for two numbers that multiply to \(-63\) and add to \(-2\). The numbers \(-9\) and \(7\) work:
\[ a^2 - 9a + 7a - 63 \]
Grouping the terms:
\[ a(a - 9) + 7(a - 9) \]
Factoring out the common term:
\[ (a - 9)(a + 7) \]

### Step 5: Substitute back the value of \( a \)
Now we substitute back \( a = 2x^2 - 3x \):
\[ (2x^2 - 3x - 9)(2x^2 - 3x + 7) \]

### Step 6: Final expression
Thus, the factors of the original expression are:
\[ (2x^2 - 3x - 9)(2x^2 - 3x + 7) \]