<s>[INST] Easy algebra question: show equality? Let $c:=(u+d)/2$ and $-1<d<0<u$.
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Can you show the steps to arrive from lhs to rhs?
 [/INST] $$
\frac{1 + u}{2}(1 + 2c) + \frac{1+d}{2}(1 + c) = \frac{1 + u}{2}(1 + c) + \frac{1 + u}{2}c + \frac{1+d}{2}(1 + c)
$$
$$
=(1 + c)\biggl(\frac{1 + u}{2} + \frac{1 + d}{2} \biggr) + \frac{1 + u}{2}c = (1 + c)\biggl(1 + \frac{u + d}{2}\biggr) + \frac{1 + u}{2}c
$$
$$
= (1 + c)^2 + \frac{1 + u}{2}c
$$
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