Question

Simplify: ${\left(4m+5n\right)}^2+{\left(5m+4n\right)}^2$.

Answer 1

Simplify the expression:

Step 1. Expand ${\left(4m+5n\right)}^2$.

Expand the term ${\left(4m+5n\right)}^2$:

$\begin{array}{c}{(4m+5n)}^2={\left(4m\right)}^2+{\left(5n\right)}^2+2\left(4m\right)\left(5n\right)\\ =16m^2+25n^2+40mn\end{array}$ $[{(\mathrm{a}+\mathrm{b})}^2={\mathrm{a}}^2+{\mathrm{b}}^2+2\mathrm{ab}]$

Step 2. Expand ${\left(5m+4n\right)}^2$.

Expand the term ${\left(5m+4n\right)}^2$:

$\begin{array}{c}{(5m+4n)}^2={\left(5m\right)}^2+{\left(4n\right)}^2+2\left(5m\right)\left(4n\right)\\ =25m^2+16n^2+40mn\end{array}$

Step 3. Add the expanded term of ${\left(4m+5n\right)}^2+{\left(5m+4n\right)}^2$.

Add the expanded terms of ${\left(4m+5n\right)}^2+{\left(5m+4n\right)}^2$:

$\begin{array}{c}{(4m+5n)}^2+{(5m+4n)}^2=[16m^2+25n^2+40mn]+[25m^2+16n^2+40mn]\\ =(16+25)m^2+(25+16)n^2+(40+40)mn\\ =41m^2+41n^2+80mn\end{array}$ $[{(\mathrm{a}+\mathrm{b})}^2={\mathrm{a}}^2+{\mathrm{b}}^2+2\mathrm{ab}]$

Hence, the value of the given expression is ${(4m+5n)}^2+{(5m+4n)}^2=41m^2+41n^2+80mn$.