Question
Divide $\left(a^2+2 a b+b^2\right)-\left(a^2+2 a c+c^2\right)$ by $(2 a+b+c)$
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Answer
$\frac{\big(\text{a}^2+2\text{ab}+\text{b}^2\big)-\big(\text{a}^2+2\text{ac}+\text{c}^2\big)}{2\text{a}+\text{b}+\text{c}}$
$=\frac{(\text{a}+\text{b}^2-(\text{a}+\text{c})^2}{2\text{a}+\text{b}+\text{c}}$
$=\frac{(\text{a}+\text{b}+\text{a}+\text{c})(\text{a}+\text{b}-\text{a}-\text{c})}{2\text{a}+\text{b}+\text{c}}$
$=\frac{(2\text{a}+\text{b}+\text{c})(\text{b}-\text{c})}{2\text{a}+\text{b}+\text{c}}=\text{b}-\text{c}$
$=\frac{(\text{a}+\text{b}^2-(\text{a}+\text{c})^2}{2\text{a}+\text{b}+\text{c}}$
$=\frac{(\text{a}+\text{b}+\text{a}+\text{c})(\text{a}+\text{b}-\text{a}-\text{c})}{2\text{a}+\text{b}+\text{c}}$
$=\frac{(2\text{a}+\text{b}+\text{c})(\text{b}-\text{c})}{2\text{a}+\text{b}+\text{c}}=\text{b}-\text{c}$
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