To multiply the expression \((6a - 7b)(9a^2 - 2ab + 7b^2)\), we will use the distributive property (also known as the FOIL method for binomials):
\[ (6a - 7b)(9a^2 - 2ab + 7b^2) = 6a(9a^2 - 2ab + 7b^2) - 7b(9a^2 - 2ab + 7b^2) \]
Now, distribute \(6a\) to each term in the second polynomial:
- \(6a \cdot 9a^2 = 54a^3\)
- \(6a \cdot (-2ab) = -12a^2b\)
- \(6a \cdot 7b^2 = 42ab^2\)
So, the first part is:
\[ 54a^3 - 12a^2b + 42ab^2 \]
Next, distribute \(-7b\) to each term in the second polynomial:
- \(-7b \cdot 9a^2 = -63a^2b\)
- \(-7b \cdot (-2ab) = 14ab^2\)
- \(-7b \cdot 7b^2 = -49b^3\)
So, the second part is:
\[ -63a^2b + 14ab^2 - 49b^3 \]
Now, combine all the terms:
\[ 54a^3 + (-12a^2b - 63a^2b) + (42ab^2 + 14ab^2) - 49b^3 \]
Combine like terms:
- The \(a^3\) term: \(54a^3\)
- The \(a^2b\) terms: \(-12a^2b - 63a^2b = -75a^2b\)
- The \(ab^2\) terms: \(42ab^2 + 14ab^2 = 56ab^2\)
- The \(b^3\) term: \(-49b^3\)
Putting it all together, the simplified result is:
\[ 54a^3 - 75a^2b + 56ab^2 - 49b^3 \]
Thus, the final answer is:
\[ \boxed{54a^3 - 75a^2b + 56ab^2 - 49b^3} \]