Which of the following is correct ? (a) `(6x + y) (x - 6y) = 6x ^(2) + 35 xy - 6y ^(2)` (b) `(6x + y) (x - 6y) = 6x ^(2) - 35 xy -6y ^(2)` (c) `(6x + y) (x - 6y) = 6x ^(2) - 37 xy - 6y ^(2)` (d) `(6x+ y) (x- 6y) = 6x ^(2) + 37 xy - 6y ^(2)`

Options

A. `(6x + y) (x - 6y) = 6x ^(2) + 35 xy - 6y ^(2)`
B. `(6x + y) (x - 6y) = 6x ^(2) - 35 xy -6y ^(2)`
C. `(6x + y) (x - 6y) = 6x ^(2) - 37 xy - 6y ^(2)`
D. `(6x+ y) (x- 6y) = 6x ^(2) + 37 xy - 6y ^(2)`

Answer

To solve the problem, we need to expand the expression \((6x + y)(x - 6y)\) and compare it with the provided options to determine which one is correct.

### Step 1: Expand the expression
We will use the distributive property (also known as the FOIL method for binomials) to expand \((6x + y)(x - 6y)\).

1. **Multiply the first terms**:
\[ 6x \cdot x = 6x^2 \]

2. **Multiply the outer terms**:
\[ 6x \cdot (-6y) = -36xy \]

3. **Multiply the inner terms**:
\[ y \cdot x = xy \]

4. **Multiply the last terms**:
\[ y \cdot (-6y) = -6y^2 \]

### Step 2: Combine all the results
Now, we will combine all the results from the multiplication:
\[ 6x^2 - 36xy + xy - 6y^2 \]

### Step 3: Simplify the expression
Combine like terms:
\[ 6x^2 + (-36xy + xy) - 6y^2 = 6x^2 - 35xy - 6y^2 \]

### Conclusion
The expanded form of \((6x + y)(x - 6y)\) is:
\[ 6x^2 - 35xy - 6y^2 \]

Now, we will compare this result with the options given:

- (a) \(6x^2 + 35xy - 6y^2\) - Incorrect
- (b) \(6x^2 - 35xy - 6y^2\) - Correct
- (c) \(6x^2 - 37xy - 6y^2\) - Incorrect
- (d) \(6x^2 + 37xy - 6y^2\) - Incorrect

Thus, the correct option is **(b)**.