To rewrite the rational expression with the indicated denominator, we want to express
\[ \frac{14m - n}{m^2 + mn + n^2} \]
as
\[ \frac{?}{n^3 - m^3}. \]
First, we know that the denominator \( n^3 - m^3 \) can be factored using the difference of cubes formula:
\[ n^3 - m^3 = (n - m)(n^2 + nm + m^2). \]
We see that the denominator in the original fraction, \( m^2 + mn + n^2 \), is actually one of the factors in the factored form of \( n^3 - m^3 \).
Thus, we can rewrite the expression:
\[ \frac{14m - n}{m^2 + mn + n^2} = \frac{14m - n}{(n - m)(n^2 + nm + m^2)}. \]
To write this with the denominator of \( n^3 - m^3 \), we multiply both the numerator and denominator of our original expression by \( n - m \):
\[ \frac{(14m - n)(n - m)}{(m^2 + mn + n^2)(n - m)} = \frac{(14m - n)(n - m)}{n^3 - m^3}. \]
Now, the numerator is \( (14m - n)(n - m) \). Therefore, the rewritten expression is:
\[ \frac{(14m - n)(n - m)}{n^3 - m^3}. \]
Thus, the expression you're looking for, represented in terms of the desired denominator, is
\[ (14m - n)(n - m). \]