Divide. If the polynomial does not divide evenly, include the remainder as a fraction \( \left(63 m^{4}-60 m^{3}-76 m^{2}+86\right) \div(7 m-9) \)

To divide the polynomial \( 63 m^{4}-60 m^{3}-76 m^{2}+86 \) by \( 7 m-9 \), we can use polynomial long division. 1. Divide the first term of the dividend \( 63 m^{4} \) by the first term of the divisor \( 7 m \) to get \( 9 m^{3} \). 2. Multiply \( 9 m^{3} \) by the entire divisor \( 7 m - 9 \) to get \( 63 m^{4} - 81 m^{3} \). 3. Subtract this result from the original polynomial: \[ (63 m^{4}-60 m^{3}-76 m^{2}+86) - (63 m^{4} - 81 m^{3}) = 21 m^{3} - 76 m^{2} + 86. \] 4. Next, divide \( 21 m^{3} \) by \( 7 m \) to get \( 3 m^{2} \). 5. Multiply \( 3 m^{2} \) by \( 7 m - 9 \) to get \( 21 m^{3} - 27 m^{2} \). 6. Perform another subtraction: \[ (21 m^{3} - 76 m^{2} + 86) - (21 m^{3} - 27 m^{2}) = -49 m^{2} + 86. \] 7. Now, divide \( -49 m^{2} \) by \( 7 m \) to get \( -7 m \). 8. Multiply \( -7 m \) by \( 7 m - 9 \) to get \( -49 m^{2} + 63 m \). 9. Subtract again: \[ (-49 m^{2} + 86) - (-49 m^{2} + 63 m) = 86 - 63 m = -63 m + 86. \] 10. Finally, divide \( -63 m \) by \( 7 m \) to get \( -9 \). 11. Multiply \( -9 \) by \( 7 m - 9 \) to get \( -63 m + 81 \). 12. Perform the last subtraction: \[ (-63 m + 86) - (-63 m + 81) = 86 - 81 = 5. \] Putting it all together: The quotient is \( 9 m^{3} + 3 m^{2} - 7 m - 9 \), and the remainder is \( 5 \). Thus, the answer is: \[ \boxed{9 m^3 + 3 m^2 - 7 m - 9 + \frac{5}{7m-9}}. \]