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$(\mathbf{a}+\mathbf{b}) \cdot(\mathbf{b}+\mathbf{c}) \times(\mathbf{a}+\mathbf{b}+\mathbf{c})$ is equal to
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$\left[\begin{array}{lll}\mathbf{a} & \mathbf{b} & \mathbf{c}]\end{array}\right.$
$\begin{aligned} & (a+b) \cdot(b+c) \times(a+b+c) \\ & =(a+b) \cdot(b \times a+0+b \times c+c \times a+c \times b+0) \\ & =(a+b) \cdot(b \times a+c \times a) \\ & =a(b \times a)+a(c \times a)+b(b \times a)+b \cdot(c+a) \\ & =0+0+0+[b c a]=[a b c]\end{aligned}$
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