Question

What should be subtracted from 3m²n + 5mn² $-$ 2m²n² to get $-$3m²n + 7mn² + 4m²n² ?

Answer 1

$$\begin{gathered} \mathrm{Let}p\left(x\right)\mathrm{be}\mathrm{subtracted}\mathrm{from}3m^{2}n+5m{n}^2-2m^2{n}^{2}toget-3m^{2}n+7m{n}^2+4m^2{n}^2. \\ \therefore \left(3m^{2}n+5m{n}^2-2m^2{n}^2\right)-p\left(x\right)=\left(-3m^{2}n+7m{n}^2+4m^2{n}^2\right) \\ \Rightarrow p\left(x\right)=\left(3m^{2}n+5m{n}^2-2m^2{n}^2\right)-\left(-3m^{2}n+7m{n}^2+4m^2{n}^2\right) \\ \Rightarrow p\left(x\right)=3m^{2}n+5m{n}^2-2m^2{n}^2+3m^{2}n-7m{n}^2-4m^2{n}^2 \\ \Rightarrow p\left(x\right)=3m^{2}n+3m^{2}n+5m{n}^2-7m{n}^2-2m^2{n}^2-4m^2{n}^2 \\ \Rightarrow p\left(x\right)=6m^{2}n-2m{n}^2-6m^2{n}^2 \\ \therefore \left(6m^{2}n-2m{n}^2-6m^2{n}^2\right)\mathrm{can}\mathrm{be}\mathrm{subtracted}\mathrm{from}\left(3m^{2}n+5m{n}^2-2m^2{n}^2\right)\mathrm{to}\mathrm{get}\left(-3m^{2}n+7m{n}^2+4m^2{n}^2\right). \end{gathered}$$