Solve (x^3+2x^2-3x-6-21/x-2)*(x-2)=

$$( { x }^{ 3 } +2 { x }^{ 2 } -3x-6- \frac{ 21 }{ x-2 } ) \times (x-2)=$$

Evaluate

$$x^{4}-7x^{2}-9$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{3}+2x^{2}-3x-6 times \frac{x-2}{x-2}.

$$\left(\frac{\left(x^{3}+2x^{2}-3x-6\right)\left(x-2\right)}{x-2}-\frac{21}{x-2}\right)\left(x-2\right)$$

Since \frac{\left(x^{3}+2x^{2}-3x-6\right)\left(x-2\right)}{x-2} and \frac{21}{x-2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{3}+2x^{2}-3x-6\right)\left(x-2\right)-21}{x-2}\left(x-2\right)$$

Do the multiplications in \left(x^{3}+2x^{2}-3x-6\right)\left(x-2\right)-21.

$$\frac{x^{4}-2x^{3}+2x^{3}-4x^{2}-3x^{2}+6x-6x+12-21}{x-2}\left(x-2\right)$$

Combine like terms in x^{4}-2x^{3}+2x^{3}-4x^{2}-3x^{2}+6x-6x+12-21.

$$\frac{x^{4}-7x^{2}-9}{x-2}\left(x-2\right)$$

Cancel out x-2 and x-2.

$$x^{4}-7x^{2}-9$$

Expand

$$x^{4}-7x^{2}-9$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{3}+2x^{2}-3x-6 times \frac{x-2}{x-2}.

$$\left(\frac{\left(x^{3}+2x^{2}-3x-6\right)\left(x-2\right)}{x-2}-\frac{21}{x-2}\right)\left(x-2\right)$$

Since \frac{\left(x^{3}+2x^{2}-3x-6\right)\left(x-2\right)}{x-2} and \frac{21}{x-2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{3}+2x^{2}-3x-6\right)\left(x-2\right)-21}{x-2}\left(x-2\right)$$

Do the multiplications in \left(x^{3}+2x^{2}-3x-6\right)\left(x-2\right)-21.

$$\frac{x^{4}-2x^{3}+2x^{3}-4x^{2}-3x^{2}+6x-6x+12-21}{x-2}\left(x-2\right)$$

Combine like terms in x^{4}-2x^{3}+2x^{3}-4x^{2}-3x^{2}+6x-6x+12-21.

$$\frac{x^{4}-7x^{2}-9}{x-2}\left(x-2\right)$$

Cancel out x-2 and x-2.

$$x^{4}-7x^{2}-9$$