Solve f(x)=3/x+1-(2x+3)(x^2-1)

$$f ( x ) = \frac { 3 } { x + 1 } - ( 2 x + 3 ) ( x ^ { 2 } - 1 )$$

Evaluate

$$\frac{\left(x+2\right)\left(3+x-x^{2}-2x^{3}\right)}{x+1}$$

Short Solution Steps

Use the distributive property to multiply 2x+3 by x^{2}-1.

$$\frac{3}{x+1}-\left(2x^{3}-2x+3x^{2}-3\right)$$

To find the opposite of 2x^{3}-2x+3x^{2}-3, find the opposite of each term.

$$\frac{3}{x+1}-2x^{3}+2x-3x^{2}+3$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -2x^{3}+2x-3x^{2}+3 times \frac{x+1}{x+1}.

$$\frac{3}{x+1}+\frac{\left(-2x^{3}+2x-3x^{2}+3\right)\left(x+1\right)}{x+1}$$

Since \frac{3}{x+1} and \frac{\left(-2x^{3}+2x-3x^{2}+3\right)\left(x+1\right)}{x+1} have the same denominator, add them by adding their numerators.

$$\frac{3+\left(-2x^{3}+2x-3x^{2}+3\right)\left(x+1\right)}{x+1}$$

Do the multiplications in 3+\left(-2x^{3}+2x-3x^{2}+3\right)\left(x+1\right).

$$\frac{3-2x^{4}-2x^{3}+2x^{2}+2x-3x^{3}-3x^{2}+3x+3}{x+1}$$

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

$$\frac{6-2x^{4}-5x^{3}-x^{2}+5x}{x+1}$$

Expand

$$\frac{6+5x-x^{2}-5x^{3}-2x^{4}}{x+1}$$

Short Solution Steps

Use the distributive property to multiply 2x+3 by x^{2}-1.

$$\frac{3}{x+1}-\left(2x^{3}-2x+3x^{2}-3\right)$$

To find the opposite of 2x^{3}-2x+3x^{2}-3, find the opposite of each term.

$$\frac{3}{x+1}-2x^{3}+2x-3x^{2}+3$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -2x^{3}+2x-3x^{2}+3 times \frac{x+1}{x+1}.

$$\frac{3}{x+1}+\frac{\left(-2x^{3}+2x-3x^{2}+3\right)\left(x+1\right)}{x+1}$$

Since \frac{3}{x+1} and \frac{\left(-2x^{3}+2x-3x^{2}+3\right)\left(x+1\right)}{x+1} have the same denominator, add them by adding their numerators.

$$\frac{3+\left(-2x^{3}+2x-3x^{2}+3\right)\left(x+1\right)}{x+1}$$

Do the multiplications in 3+\left(-2x^{3}+2x-3x^{2}+3\right)\left(x+1\right).

$$\frac{3-2x^{4}-2x^{3}+2x^{2}+2x-3x^{3}-3x^{2}+3x+3}{x+1}$$

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

$$\frac{6-2x^{4}-5x^{3}-x^{2}+5x}{x+1}$$