Solve 1+5/x^2-frac{3{x+2}}

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Divide 5 by \frac{x^{3}+2x^{2}-3}{x+2} by multiplying 5 by the reciprocal of \frac{x^{3}+2x^{2}-3}{x+2}.

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

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

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

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

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

Since \frac{\left(x-1\right)\left(x^{2}+3x+3\right)}{\left(x-1\right)\left(x^{2}+3x+3\right)} and \frac{5\left(x+2\right)}{\left(x-1\right)\left(x^{2}+3x+3\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x-1\right)\left(x^{2}+3x+3\right)+5\left(x+2\right).

$$\frac{x^{3}+3x^{2}+3x-x^{2}-3x-3+5x+10}{\left(x-1\right)\left(x^{2}+3x+3\right)}$$

Combine like terms in x^{3}+3x^{2}+3x-x^{2}-3x-3+5x+10.

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

Expand \left(x-1\right)\left(x^{2}+3x+3\right).

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

Differentiate w.r.t. x

$$-\frac{5\left(2x^{3}+8x^{2}+8x+3\right)}{\left(x-1\right)^{2}\left(x^{4}+6x^{3}+15x^{2}+18x+9\right)}$$