Solve 2/2x^2+5x+3-1/2x^2-x-6+3/x^2-x-2

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

Evaluate

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

Short Solution Steps

Factor 2x^{2}+5x+3. Factor 2x^{2}-x-6.

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

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

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

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

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

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

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

Combine like terms in 2x-4-x-1.

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

Factor x^{2}-x-2.

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

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

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

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

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

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

$$\frac{x-5+6x+9}{\left(x-2\right)\left(x+1\right)\left(2x+3\right)}$$

Combine like terms in x-5+6x+9.

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

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

$$\frac{7x+4}{2x^{3}+x^{2}-7x-6}$$

Differentiate w.r.t. x

$$-\frac{28x^{3}+31x^{2}+8x+14}{\left(\left(x-2\right)\left(x+1\right)\left(2x+3\right)\right)^{2}}$$