The rational expression
`((x^2 - xy - 12 y^2)(x^2 - xy - 12y^2))/((x^2 - 16y^2)(x^2 - 9y^2))` when simplified equals.

Options

A. `1 `
B. `xy`
C. `(x + y) `
D. `(x - y) `

Answer

To simplify the rational expression

\[ \frac{(x^2 - xy - 12y^2)(x^2 - xy - 12y^2)}{(x^2 - 16y^2)(x^2 - 9y^2)}, \]

we will follow these steps:

### Step 1: Factor the Denominator

1. **Factor \(x^2 - 16y^2\)**:
- Recognize this as a difference of squares: \(x^2 - (4y)^2\).
- This can be factored as \((x - 4y)(x + 4y)\).

2. **Factor \(x^2 - 9y^2\)**:
- Again, this is a difference of squares: \(x^2 - (3y)^2\).
- This can be factored as \((x - 3y)(x + 3y)\).

Thus, the denominator becomes:
\[ (x - 4y)(x + 4y)(x - 3y)(x + 3y). \]

### Step 2: Factor the Numerator

1. **Factor \(x^2 - xy - 12y^2\)**:
- We need to find two numbers that multiply to \(-12\) (the product of \(-12\) and \(1\)) and add up to \(-1\) (the coefficient of \(xy\)).
- The numbers \(-4\) and \(3\) work since \(-4 \cdot 3 = -12\) and \(-4 + 3 = -1\).
- Rewrite the expression:
\[ x^2 - 4xy + 3xy - 12y^2 = (x^2 - 4xy) + (3xy - 12y^2). \]
- Factor by grouping:
\[ x(x - 4y) + 3y(x - 4y) = (x + 3y)(x - 4y). \]

Thus, the numerator becomes:
\[ ((x + 3y)(x - 4y))^2 = (x + 3y)(x - 4y)(x + 3y)(x - 4y). \]

### Step 3: Substitute Back into the Expression

Now we can substitute the factored forms back into the original expression:
\[ \frac{(x + 3y)(x - 4y)(x + 3y)(x - 4y)}{(x - 4y)(x + 4y)(x - 3y)(x + 3y)}. \]

### Step 4: Cancel Common Factors

1. Cancel one \((x + 3y)\) from the numerator with one \((x + 3y)\) from the denominator.
2. Cancel one \((x - 4y)\) from the numerator with one \((x - 4y)\) from the denominator.

After cancellation, we are left with:
\[ \frac{(x + 3y)(x - 4y)}{(x - 3y)(x + 4y)}. \]

### Final Expression

Thus, the simplified expression is:
\[ \frac{(x + 3y)(x - 4y)}{(x - 3y)(x + 4y)}. \]

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