1. **State the problem:** Given that $k = m + n$ and $m = \frac{3}{4}n$, express $\frac{7k}{5 - m}$ in the form $\frac{a + bn^2}{cn - d}$ where $a$, $b$, and $c$ are integers and $d$ is prime. 2. **Substitute $m$ into $k$:** $$k = m + n = \frac{3}{4}n + n = \frac{3}{4}n + \frac{4}{4}n = \frac{7}{4}n$$ 3. **Rewrite the expression:** $$\frac{7k}{5 - m} = \frac{7 \times \frac{7}{4}n}{5 - \frac{3}{4}n} = \frac{\frac{49}{4}n}{5 - \frac{3}{4}n}$$ 4. **Clear denominators in the denominator:** Multiply numerator and denominator by 4 to eliminate fractions: $$\frac{\frac{49}{4}n \times 4}{(5 - \frac{3}{4}n) \times 4} = \frac{49n}{20 - 3n}$$ 5. **Rewrite denominator to match form $cn - d$:** $$20 - 3n = -(3n - 20)$$ 6. **Multiply numerator and denominator by $-1$ to get positive leading coefficient in denominator:** $$\frac{49n}{20 - 3n} = \frac{-49n}{3n - 20}$$ 7. **Final form:** $$\frac{a + bn^2}{cn - d} = \frac{0 - 49n^2}{3n - 20}$$ Here, $a = 0$, $b = -49$, $c = 3$, and $d = 20$. However, $d$ must be prime, and 20 is not prime. 8. **Check original problem's form:** The user wants $\frac{a + bn^2}{cn - d}$ with $a,b,c$ integers and $d$ prime. 9. **Re-express $k$ correctly:** The user's initial substitution was incorrect. Given $m = \frac{3}{4}n$, then $$k = m + n = \frac{3}{4}n + n = \frac{3}{4}n + \frac{4}{4}n = \frac{7}{4}n$$ 10. **Calculate numerator:** $$7k = 7 \times \frac{7}{4}n = \frac{49}{4}n$$ 11. **Calculate denominator:** $$5 - m = 5 - \frac{3}{4}n = \frac{20}{4} - \frac{3}{4}n = \frac{20 - 3n}{4}$$ 12. **Form the fraction:** $$\frac{7k}{5 - m} = \frac{\frac{49}{4}n}{\frac{20 - 3n}{4}} = \frac{49}{4}n \times \frac{4}{20 - 3n} = \frac{49n}{20 - 3n}$$ 13. **Rewrite denominator as $cn - d$:** $$20 - 3n = -(3n - 20)$$ 14. **Multiply numerator and denominator by $-1$ to get positive $c$:** $$\frac{49n}{20 - 3n} = \frac{-49n}{3n - 20}$$ 15. **Express numerator as $a + bn^2$:** Since numerator is $-49n$, write as $0 + (-49)n^2$ only if $n$ is squared, but here it is linear in $n$. 16. **Conclusion:** The expression cannot be written exactly in the form $\frac{a + bn^2}{cn - d}$ with $a,b,c$ integers and $d$ prime because numerator is linear in $n$, not quadratic. **Final answer:** $$\frac{7k}{5 - m} = \frac{49n}{20 - 3n} = \frac{-49n}{3n - 20}$$ where $a=0$, $b=0$, $c=3$, and $d=20$ (not prime).