Multiply using both the horizontal and the vertical methods, and hence verify that they give the same result.
$(2 a^{3} + 3 b) (6 a - 4 b^{2})$

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Solution

$\begin{matrix} (2 a^{3} + 3 b) (6 a - 4 b^{2}) \Rightarrow 12 a^{4} - 8 a^{3} b^{2} + 18 a b - 12 b^{3} 2 a^{3} - + 3 b \times 6 a - 4 b^{2} ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ - 8 a^{3} b^{2} - 12 b^{3} 12 a^{4} + 18 a b 12 a^{4} - 8 a^{3} b^{2} + 18 a b - 12 b^{3} \end{matrix}$

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