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Question
Simplify:
(5x + 3)(x − 1)(3x − 2)
Solution
To simplify, we will proceed as follows:
\[\left( 5x + 3 \right)\left( x - 1 \right)\left( 3x - 2 \right)\]
\[ = \left[ \left( 5x + 3 \right)\left( x - 1 \right) \right]\left( 3x - 2 \right)\]
\[= \left[ 5x\left( x - 1 \right) + 3\left( x - 1 \right) \right]\left( 3x - 2 \right)\] (Distributive law)
\[= \left[ 5 x^2 - 5x + 3x - 3 \right]\left( 3x - 2 \right)\]
\[ = \left[ 5 x^2 - 2x - 3 \right]\left( 3x - 2 \right)\]
\[ = 3x\left( 5 x^2 - 2x - 3 \right) - 2\left( 5 x^2 - 2x - 3 \right)\]
\[ = 15 x^3 - 6 x^2 - 9x - \left[ 10 x^2 - 4x - 6 \right]\]
\[ = 15 x^3 - 6 x^2 - 9x - 10 x^2 + 4x + 6\]
\[= 15 x^3 - 6 x^2 - 10 x^2 - 9x + 4x + 6\] (Rearranging)
\[= 15 x^3 - 16 x^2 - 5x + 6\] (Combining like terms)
Thus, the answer is \[15 x^3 - 16 x^2 - 5x + 6\].
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