Solve x^2+4x+3+3/x^2-1-2/x+3

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

Evaluate

$$\frac{x^{5}+7x^{4}+14x^{3}-12x+2}{\left(x+3\right)\left(x^{2}-1\right)}$$

Short Solution Steps

Factor x^{2}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2}+4x+3 times \frac{\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}.

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

Since \frac{\left(x^{2}+4x+3\right)\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)} and \frac{3}{\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x^{2}+4x+3\right)\left(x-1\right)\left(x+1\right)+3.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-1\right)\left(x+1\right) and x+3 is \left(x-1\right)\left(x+1\right)\left(x+3\right). Multiply \frac{x^{4}+2x^{2}+4x^{3}-4x}{\left(x-1\right)\left(x+1\right)} times \frac{x+3}{x+3}. Multiply \frac{2}{x+3} times \frac{\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}.

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

Since \frac{\left(x^{4}+2x^{2}+4x^{3}-4x\right)\left(x+3\right)}{\left(x-1\right)\left(x+1\right)\left(x+3\right)} and \frac{2\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x^{4}+2x^{2}+4x^{3}-4x\right)\left(x+3\right)-2\left(x-1\right)\left(x+1\right).

$$\frac{x^{5}+3x^{4}+2x^{3}+6x^{2}+4x^{4}+12x^{3}-4x^{2}-12x-2x^{2}-2x+2x+2}{\left(x-1\right)\left(x+1\right)\left(x+3\right)}$$

Combine like terms in x^{5}+3x^{4}+2x^{3}+6x^{2}+4x^{4}+12x^{3}-4x^{2}-12x-2x^{2}-2x+2x+2.

$$\frac{x^{5}+7x^{4}+14x^{3}-12x+2}{\left(x-1\right)\left(x+1\right)\left(x+3\right)}$$

Expand \left(x-1\right)\left(x+1\right)\left(x+3\right).

$$\frac{x^{5}+7x^{4}+14x^{3}-12x+2}{x^{3}+3x^{2}-x-3}$$

Differentiate w.r.t. x

$$\frac{2\left(x^{7}+8x^{6}+19x^{5}+3x^{4}-44x^{3}-48x^{2}-6x+19\right)}{\left(\left(x+3\right)\left(x^{2}-1\right)\right)^{2}}$$