Solve 2x-9+26x+24/x^2+3x+2-2/x

$$2 x - 9 + \frac { 26 x + 24 } { x ^ { 2 } + 3 x + 2 } - \frac { 2 } { x }$$

Evaluate

$$\frac{2x^{4}-3x^{3}+x^{2}-4}{x\left(x+1\right)\left(x+2\right)}$$

Short Solution Steps

Factor x^{2}+3x+2.

$$2x-9+\frac{26x+24}{\left(x+1\right)\left(x+2\right)}-\frac{2}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2x-9 times \frac{\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}.

$$\frac{\left(2x-9\right)\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}+\frac{26x+24}{\left(x+1\right)\left(x+2\right)}-\frac{2}{x}$$

Since \frac{\left(2x-9\right)\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)} and \frac{26x+24}{\left(x+1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(2x-9\right)\left(x+1\right)\left(x+2\right)+26x+24}{\left(x+1\right)\left(x+2\right)}-\frac{2}{x}$$

Do the multiplications in \left(2x-9\right)\left(x+1\right)\left(x+2\right)+26x+24.

$$\frac{2x^{3}+6x^{2}+4x-9x^{2}-27x-18+26x+24}{\left(x+1\right)\left(x+2\right)}-\frac{2}{x}$$

Combine like terms in 2x^{3}+6x^{2}+4x-9x^{2}-27x-18+26x+24.

$$\frac{2x^{3}-3x^{2}+3x+6}{\left(x+1\right)\left(x+2\right)}-\frac{2}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+1\right)\left(x+2\right) and x is x\left(x+1\right)\left(x+2\right). Multiply \frac{2x^{3}-3x^{2}+3x+6}{\left(x+1\right)\left(x+2\right)} times \frac{x}{x}. Multiply \frac{2}{x} times \frac{\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}.

$$\frac{\left(2x^{3}-3x^{2}+3x+6\right)x}{x\left(x+1\right)\left(x+2\right)}-\frac{2\left(x+1\right)\left(x+2\right)}{x\left(x+1\right)\left(x+2\right)}$$

Since \frac{\left(2x^{3}-3x^{2}+3x+6\right)x}{x\left(x+1\right)\left(x+2\right)} and \frac{2\left(x+1\right)\left(x+2\right)}{x\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2x^{3}-3x^{2}+3x+6\right)x-2\left(x+1\right)\left(x+2\right)}{x\left(x+1\right)\left(x+2\right)}$$

Do the multiplications in \left(2x^{3}-3x^{2}+3x+6\right)x-2\left(x+1\right)\left(x+2\right).

$$\frac{2x^{4}-3x^{3}+3x^{2}+6x-2x^{2}-4x-2x-4}{x\left(x+1\right)\left(x+2\right)}$$

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

$$\frac{2x^{4}-3x^{3}+x^{2}-4}{x\left(x+1\right)\left(x+2\right)}$$

Expand x\left(x+1\right)\left(x+2\right).

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

Expand

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

Short Solution Steps

Factor x^{2}+3x+2.

$$2x-9+\frac{26x+24}{\left(x+1\right)\left(x+2\right)}-\frac{2}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2x-9 times \frac{\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}.

$$\frac{\left(2x-9\right)\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}+\frac{26x+24}{\left(x+1\right)\left(x+2\right)}-\frac{2}{x}$$

Since \frac{\left(2x-9\right)\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)} and \frac{26x+24}{\left(x+1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(2x-9\right)\left(x+1\right)\left(x+2\right)+26x+24}{\left(x+1\right)\left(x+2\right)}-\frac{2}{x}$$

Do the multiplications in \left(2x-9\right)\left(x+1\right)\left(x+2\right)+26x+24.

$$\frac{2x^{3}+6x^{2}+4x-9x^{2}-27x-18+26x+24}{\left(x+1\right)\left(x+2\right)}-\frac{2}{x}$$

Combine like terms in 2x^{3}+6x^{2}+4x-9x^{2}-27x-18+26x+24.

$$\frac{2x^{3}-3x^{2}+3x+6}{\left(x+1\right)\left(x+2\right)}-\frac{2}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+1\right)\left(x+2\right) and x is x\left(x+1\right)\left(x+2\right). Multiply \frac{2x^{3}-3x^{2}+3x+6}{\left(x+1\right)\left(x+2\right)} times \frac{x}{x}. Multiply \frac{2}{x} times \frac{\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}.

$$\frac{\left(2x^{3}-3x^{2}+3x+6\right)x}{x\left(x+1\right)\left(x+2\right)}-\frac{2\left(x+1\right)\left(x+2\right)}{x\left(x+1\right)\left(x+2\right)}$$

Since \frac{\left(2x^{3}-3x^{2}+3x+6\right)x}{x\left(x+1\right)\left(x+2\right)} and \frac{2\left(x+1\right)\left(x+2\right)}{x\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2x^{3}-3x^{2}+3x+6\right)x-2\left(x+1\right)\left(x+2\right)}{x\left(x+1\right)\left(x+2\right)}$$

Do the multiplications in \left(2x^{3}-3x^{2}+3x+6\right)x-2\left(x+1\right)\left(x+2\right).

$$\frac{2x^{4}-3x^{3}+3x^{2}+6x-2x^{2}-4x-2x-4}{x\left(x+1\right)\left(x+2\right)}$$

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

$$\frac{2x^{4}-3x^{3}+x^{2}-4}{x\left(x+1\right)\left(x+2\right)}$$

Expand x\left(x+1\right)\left(x+2\right).

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