Solve 4/3x+2+14/x^2+1+11/(x^2+1)^2

$$\frac { 4 } { 3 x + 2 } + \frac { 14 } { x ^ { 2 } + 1 } + \frac { 11 } { ( x ^ { 2 } + 1 ) ^ { 2 } }$$

Evaluate

$$\frac{4x^{4}+42x^{3}+36x^{2}+75x+54}{\left(3x+2\right)\left(x^{2}+1\right)^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3x+2 and x^{2}+1 is \left(3x+2\right)\left(x^{2}+1\right). Multiply \frac{4}{3x+2} times \frac{x^{2}+1}{x^{2}+1}. Multiply \frac{14}{x^{2}+1} times \frac{3x+2}{3x+2}.

$$\frac{4\left(x^{2}+1\right)}{\left(3x+2\right)\left(x^{2}+1\right)}+\frac{14\left(3x+2\right)}{\left(3x+2\right)\left(x^{2}+1\right)}+\frac{11}{\left(x^{2}+1\right)^{2}}$$

Since \frac{4\left(x^{2}+1\right)}{\left(3x+2\right)\left(x^{2}+1\right)} and \frac{14\left(3x+2\right)}{\left(3x+2\right)\left(x^{2}+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{4\left(x^{2}+1\right)+14\left(3x+2\right)}{\left(3x+2\right)\left(x^{2}+1\right)}+\frac{11}{\left(x^{2}+1\right)^{2}}$$

Do the multiplications in 4\left(x^{2}+1\right)+14\left(3x+2\right).

$$\frac{4x^{2}+4+42x+28}{\left(3x+2\right)\left(x^{2}+1\right)}+\frac{11}{\left(x^{2}+1\right)^{2}}$$

Combine like terms in 4x^{2}+4+42x+28.

$$\frac{4x^{2}+32+42x}{\left(3x+2\right)\left(x^{2}+1\right)}+\frac{11}{\left(x^{2}+1\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(3x+2\right)\left(x^{2}+1\right) and \left(x^{2}+1\right)^{2} is \left(3x+2\right)\left(x^{2}+1\right)^{2}. Multiply \frac{4x^{2}+32+42x}{\left(3x+2\right)\left(x^{2}+1\right)} times \frac{x^{2}+1}{x^{2}+1}. Multiply \frac{11}{\left(x^{2}+1\right)^{2}} times \frac{3x+2}{3x+2}.

$$\frac{\left(4x^{2}+32+42x\right)\left(x^{2}+1\right)}{\left(3x+2\right)\left(x^{2}+1\right)^{2}}+\frac{11\left(3x+2\right)}{\left(3x+2\right)\left(x^{2}+1\right)^{2}}$$

Since \frac{\left(4x^{2}+32+42x\right)\left(x^{2}+1\right)}{\left(3x+2\right)\left(x^{2}+1\right)^{2}} and \frac{11\left(3x+2\right)}{\left(3x+2\right)\left(x^{2}+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(4x^{2}+32+42x\right)\left(x^{2}+1\right)+11\left(3x+2\right)}{\left(3x+2\right)\left(x^{2}+1\right)^{2}}$$

Do the multiplications in \left(4x^{2}+32+42x\right)\left(x^{2}+1\right)+11\left(3x+2\right).

$$\frac{4x^{4}+4x^{2}+32x^{2}+32+42x^{3}+42x+33x+22}{\left(3x+2\right)\left(x^{2}+1\right)^{2}}$$

Combine like terms in 4x^{4}+4x^{2}+32x^{2}+32+42x^{3}+42x+33x+22.

$$\frac{4x^{4}+36x^{2}+54+42x^{3}+75x}{\left(3x+2\right)\left(x^{2}+1\right)^{2}}$$

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

$$\frac{4x^{4}+36x^{2}+54+42x^{3}+75x}{3x^{5}+2x^{4}+6x^{3}+4x^{2}+3x+2}$$

Expand

$$\frac{4x^{4}+42x^{3}+36x^{2}+75x+54}{\left(3x+2\right)\left(x^{2}+1\right)^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3x+2 and x^{2}+1 is \left(3x+2\right)\left(x^{2}+1\right). Multiply \frac{4}{3x+2} times \frac{x^{2}+1}{x^{2}+1}. Multiply \frac{14}{x^{2}+1} times \frac{3x+2}{3x+2}.

$$\frac{4\left(x^{2}+1\right)}{\left(3x+2\right)\left(x^{2}+1\right)}+\frac{14\left(3x+2\right)}{\left(3x+2\right)\left(x^{2}+1\right)}+\frac{11}{\left(x^{2}+1\right)^{2}}$$

Since \frac{4\left(x^{2}+1\right)}{\left(3x+2\right)\left(x^{2}+1\right)} and \frac{14\left(3x+2\right)}{\left(3x+2\right)\left(x^{2}+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{4\left(x^{2}+1\right)+14\left(3x+2\right)}{\left(3x+2\right)\left(x^{2}+1\right)}+\frac{11}{\left(x^{2}+1\right)^{2}}$$

Do the multiplications in 4\left(x^{2}+1\right)+14\left(3x+2\right).

$$\frac{4x^{2}+4+42x+28}{\left(3x+2\right)\left(x^{2}+1\right)}+\frac{11}{\left(x^{2}+1\right)^{2}}$$

Combine like terms in 4x^{2}+4+42x+28.

$$\frac{4x^{2}+32+42x}{\left(3x+2\right)\left(x^{2}+1\right)}+\frac{11}{\left(x^{2}+1\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(3x+2\right)\left(x^{2}+1\right) and \left(x^{2}+1\right)^{2} is \left(3x+2\right)\left(x^{2}+1\right)^{2}. Multiply \frac{4x^{2}+32+42x}{\left(3x+2\right)\left(x^{2}+1\right)} times \frac{x^{2}+1}{x^{2}+1}. Multiply \frac{11}{\left(x^{2}+1\right)^{2}} times \frac{3x+2}{3x+2}.

$$\frac{\left(4x^{2}+32+42x\right)\left(x^{2}+1\right)}{\left(3x+2\right)\left(x^{2}+1\right)^{2}}+\frac{11\left(3x+2\right)}{\left(3x+2\right)\left(x^{2}+1\right)^{2}}$$

Since \frac{\left(4x^{2}+32+42x\right)\left(x^{2}+1\right)}{\left(3x+2\right)\left(x^{2}+1\right)^{2}} and \frac{11\left(3x+2\right)}{\left(3x+2\right)\left(x^{2}+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(4x^{2}+32+42x\right)\left(x^{2}+1\right)+11\left(3x+2\right)}{\left(3x+2\right)\left(x^{2}+1\right)^{2}}$$

Do the multiplications in \left(4x^{2}+32+42x\right)\left(x^{2}+1\right)+11\left(3x+2\right).

$$\frac{4x^{4}+4x^{2}+32x^{2}+32+42x^{3}+42x+33x+22}{\left(3x+2\right)\left(x^{2}+1\right)^{2}}$$

Combine like terms in 4x^{4}+4x^{2}+32x^{2}+32+42x^{3}+42x+33x+22.

$$\frac{4x^{4}+36x^{2}+54+42x^{3}+75x}{\left(3x+2\right)\left(x^{2}+1\right)^{2}}$$

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

$$\frac{4x^{4}+36x^{2}+54+42x^{3}+75x}{3x^{5}+2x^{4}+6x^{3}+4x^{2}+3x+2}$$