If $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are vectors such that $\mathbf{a} \cdot \mathbf{b} = -3,$ $\mathbf{a} \cdot \mathbf{c} = 4,$ and $\mathbf{b} \cdot \mathbf{c} = 6,$ then find
\[\mathbf{b} \cdot (7 \mathbf{c} - 2 \mathbf{a}).\]
That means I can write $\mathbf{b} \cdot (7 \mathbf{c} - 2 \mathbf{a})$ as $7 (\mathbf{b} \cdot \mathbf{c}) - 2 (\mathbf{b} \cdot \mathbf{a}).$ Now I can plug in the given values of the dot products and simplify. I get $7 (6) - 2 (-3) = 42 + 6 = 48.$

