Question

Simplify:
(x² − 3x + 2)(5x − 2) − (3x² + 4x − 5)(2x − 1)

Answer 1

To simplify, we will  proceed as follows:

\[\left( x^2 - 3x + 2 \right)\left( 5x - 2 \right) - \left( 3 x^2 + 4x - 5 \right)\left( 2x - 1 \right)\]

\[ = \left[ \left( x^2 - 3x + 2 \right)\left( 5x - 2 \right) \right] - \left[ \left( 3 x^2 + 4x - 5 \right)\left( 2x - 1 \right) \right]\]

\[= \left[ 5x\left( x^2 - 3x + 2 \right) - 2\left( x^2 - 3x + 2 \right) \right] - \left[ 2x\left( 3 x^2 + 4x - 5 \right) - 1 \times \left( 3 x^2 + 4x - 5 \right) \right]\]          (Distributive law)

\[= \left[ 5 x^3 - 15 x^2 + 10x - \left( 2 x^2 - 6x + 4 \right) \right] - \left[ 6 x^3 + 8 x^2 - 10x - 3 x^2 - 4x + 5 \right]\]

\[ = \left[ 5 x^3 - 15 x^2 + 10x - 2 x^2 + 6x - 4 \right] - \left[ 6 x^3 + 8 x^2 - 10x - 3 x^2 - 4x + 5 \right]\]

\[ = 5 x^3 - 15 x^2 + 10x - 2 x^2 + 6x - 4 - 6 x^3 - 8 x^2 + 10x + 3 x^2 + 4x - 5\]

\[= 5 x^3 - 6 x^3 - 15 x^2 - 2 x^2 - 8 x^2 + 3 x^2 + 10x + 6x + 10x + 4x - 5 - 4\]    (Rearranging)

\[= - x^3 - 22 x^2 + 30x - 9\]         (Combining like terms)

Thus, the answer is \[- x^3 - 22 x^2 + 30x - 9\].