2.75 Find the coefficients $m$ and $n$ of the quadratic trinomial $x^{2}+m x+n$, if it is known that the remainders when dividing by the binomials $\boldsymbol{x}-\boldsymbol{m}$ and $\boldsymbol{x}-\boldsymbol{n}$ are respectively $\boldsymbol{m}$ and $\boldsymbol{n}$.
2.75 Performing the division, we get:

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-074.jpg?height=244&width=902&top_left_y=451&top_left_x=168)

According to the condition, $2 m^{2}+n=m, m n+n^{2}+n=n$. If $n=0$, then $2 m^{2}=m$, i.e., $m=0$ or $m=0.5$. If $n \neq 0$, then $n=-m, m(m-1)=0$, i.e., $m=1, n=-1$.

Answer: $\quad m=0, n=0 ; m=0.5, n=0 ; m=1, n=-1$.