$$(w^2m+k+m\dfrac{g}{l})(w^2n+k+n\dfrac{g}{l})=\dfrac{l^2mnw^4+2glmnw^2+kl^2mw^2+kl^2nw^2+g^2mn+gklm+gkln+k^2l^2}{l^2}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(w^2m+k+m\frac{g}{l})(w^2n+k+n\frac{g}{l})& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(w^2m+k+\frac{gm}{l})(w^2n+k+\frac{gn}{l}) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{lmw^2+gm+kl}{l}\frac{lnw^2+gn+kl}{l} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{l^2mnw^4+2glmnw^2+kl^2mw^2+kl^2nw^2+g^2mn+gklm+gkln+k^2l^2}{l^2}\end{aligned} $$ | |
| ① | Multiply $m$ by $ \dfrac{g}{l} $ to get $ \dfrac{ gm }{ l } $. Step 1: Write $ m $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} m \cdot \frac{g}{l} & \xlongequal{\text{Step 1}} \frac{m}{\color{red}{1}} \cdot \frac{g}{l} \xlongequal{\text{Step 2}} \frac{ m \cdot g }{ 1 \cdot l } \xlongequal{\text{Step 3}} \frac{ gm }{ l } \end{aligned} $$ |
| ② | Multiply $n$ by $ \dfrac{g}{l} $ to get $ \dfrac{ gn }{ l } $. Step 1: Write $ n $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} n \cdot \frac{g}{l} & \xlongequal{\text{Step 1}} \frac{n}{\color{red}{1}} \cdot \frac{g}{l} \xlongequal{\text{Step 2}} \frac{ n \cdot g }{ 1 \cdot l } \xlongequal{\text{Step 3}} \frac{ gn }{ l } \end{aligned} $$ |
| ③ | Add $mw^2+k$ and $ \dfrac{gm}{l} $ to get $ \dfrac{ \color{purple}{ lmw^2+gm+kl } }{ l }$. Step 1: Write $ mw^2+k $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: To add raitonal expressions, both fractions must have the same denominator. |
| ④ | Add $nw^2+k$ and $ \dfrac{gn}{l} $ to get $ \dfrac{ \color{purple}{ lnw^2+gn+kl } }{ l }$. Step 1: Write $ nw^2+k $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: To add raitonal expressions, both fractions must have the same denominator. |
| ⑤ | Multiply $ \dfrac{lmw^2+gm+kl}{l} $ by $ \dfrac{lnw^2+gn+kl}{l} $ to get $ \dfrac{l^2mnw^4+2glmnw^2+kl^2mw^2+kl^2nw^2+g^2mn+gklm+gkln+k^2l^2}{l^2} $. Step 1: Multiply numerators and denominators. Step 2: Simplify numerator and denominator. $$ \begin{aligned} \frac{lmw^2+gm+kl}{l} \cdot \frac{lnw^2+gn+kl}{l} & \xlongequal{\text{Step 1}} \frac{ \left( lmw^2+gm+kl \right) \cdot \left( lnw^2+gn+kl \right) }{ l \cdot l } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ l^2mnw^4+glmnw^2+kl^2mw^2+glmnw^2+g^2mn+gklm+kl^2nw^2+gkln+k^2l^2 }{ l^2 } = \frac{l^2mnw^4+2glmnw^2+kl^2mw^2+kl^2nw^2+g^2mn+gklm+gkln+k^2l^2}{l^2} \end{aligned} $$ |