Use the FOIL method to multiply the binomials.

To multiply the binomials \((x-7y)(7x+6y)\) using the FOIL method, we follow these steps:

1. **First**: Multiply the first terms of each binomial.
\[
x \cdot 7x = 7x^2
\]

2. **Outer**: Multiply the outer terms of the binomials.
\[
x \cdot 6y = 6xy
\]

3. **Inner**: Multiply the inner terms of the binomials.
\[
-7y \cdot 7x = -49xy
\]

4. **Last**: Multiply the last terms of each binomial.
\[
-7y \cdot 6y = -42y^2
\]

Now, combine all these results:
\[
7x^2 + 6xy - 49xy - 42y^2
\]

Next, combine the like terms (\(6xy\) and \(-49xy\)):
\[
7x^2 + (6xy - 49xy) - 42y^2 = 7x^2 - 43xy - 42y^2
\]

Thus, the simplified result is:
\[
\boxed{7x^2 - 43xy - 42y^2}
\]