Solve (2x-3+1/x+3)+(x+1-3/2x+1)

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

Evaluate

$$\frac{6x^{3}+17x^{2}-6x-14}{\left(x+3\right)\left(2x+1\right)}$$

Short Solution Steps

Combine 2x and x to get 3x.

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

Add -3 and 1 to get -2.

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

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

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

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

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

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

$$\frac{3x^{2}+9x-2x-6+1}{x+3}-\frac{3}{2x+1}$$

Combine like terms in 3x^{2}+9x-2x-6+1.

$$\frac{3x^{2}+7x-5}{x+3}-\frac{3}{2x+1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+3 and 2x+1 is \left(x+3\right)\left(2x+1\right). Multiply \frac{3x^{2}+7x-5}{x+3} times \frac{2x+1}{2x+1}. Multiply \frac{3}{2x+1} times \frac{x+3}{x+3}.

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

Since \frac{\left(3x^{2}+7x-5\right)\left(2x+1\right)}{\left(x+3\right)\left(2x+1\right)} and \frac{3\left(x+3\right)}{\left(x+3\right)\left(2x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(3x^{2}+7x-5\right)\left(2x+1\right)-3\left(x+3\right).

$$\frac{6x^{3}+3x^{2}+14x^{2}+7x-10x-5-3x-9}{\left(x+3\right)\left(2x+1\right)}$$

Combine like terms in 6x^{3}+3x^{2}+14x^{2}+7x-10x-5-3x-9.

$$\frac{6x^{3}+17x^{2}-6x-14}{\left(x+3\right)\left(2x+1\right)}$$

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

$$\frac{6x^{3}+17x^{2}-6x-14}{2x^{2}+7x+3}$$

Differentiate w.r.t. x

$$\frac{12x^{4}+84x^{3}+185x^{2}+158x+80}{\left(\left(x+3\right)\left(2x+1\right)\right)^{2}}$$