Solve (x-5)/x^2+14x+33+4/x^2-14+40

$$\frac { ( x - 5 ) } { x ^ { 2 } + 14 x + 33 } + \frac { 4 } { x ^ { 2 } - 14 + 40 }$$

Evaluate

$$\frac{x^{3}-x^{2}+82x+2}{\left(x+3\right)\left(x+11\right)\left(x^{2}+26\right)}$$

Short Solution Steps

Add -14 and 40 to get 26.

$$\frac{x-5}{x^{2}+14x+33}+\frac{4}{x^{2}+26}$$

Factor x^{2}+14x+33.

$$\frac{x-5}{\left(x+3\right)\left(x+11\right)}+\frac{4}{x^{2}+26}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+3\right)\left(x+11\right) and x^{2}+26 is \left(x+3\right)\left(x+11\right)\left(x^{2}+26\right). Multiply \frac{x-5}{\left(x+3\right)\left(x+11\right)} times \frac{x^{2}+26}{x^{2}+26}. Multiply \frac{4}{x^{2}+26} times \frac{\left(x+3\right)\left(x+11\right)}{\left(x+3\right)\left(x+11\right)}.

$$\frac{\left(x-5\right)\left(x^{2}+26\right)}{\left(x+3\right)\left(x+11\right)\left(x^{2}+26\right)}+\frac{4\left(x+3\right)\left(x+11\right)}{\left(x+3\right)\left(x+11\right)\left(x^{2}+26\right)}$$

Since \frac{\left(x-5\right)\left(x^{2}+26\right)}{\left(x+3\right)\left(x+11\right)\left(x^{2}+26\right)} and \frac{4\left(x+3\right)\left(x+11\right)}{\left(x+3\right)\left(x+11\right)\left(x^{2}+26\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x-5\right)\left(x^{2}+26\right)+4\left(x+3\right)\left(x+11\right)}{\left(x+3\right)\left(x+11\right)\left(x^{2}+26\right)}$$

Do the multiplications in \left(x-5\right)\left(x^{2}+26\right)+4\left(x+3\right)\left(x+11\right).

$$\frac{x^{3}+26x-5x^{2}-130+4x^{2}+44x+12x+132}{\left(x+3\right)\left(x+11\right)\left(x^{2}+26\right)}$$

Combine like terms in x^{3}+26x-5x^{2}-130+4x^{2}+44x+12x+132.

$$\frac{x^{3}+82x-x^{2}+2}{\left(x+3\right)\left(x+11\right)\left(x^{2}+26\right)}$$

Expand \left(x+3\right)\left(x+11\right)\left(x^{2}+26\right).

$$\frac{x^{3}+82x-x^{2}+2}{x^{4}+14x^{3}+59x^{2}+364x+858}$$

Expand

$$\frac{x^{3}-x^{2}+82x+2}{\left(x^{2}+26\right)\left(x^{2}+14x+33\right)}$$

Short Solution Steps

Add -14 and 40 to get 26.

$$\frac{x-5}{x^{2}+14x+33}+\frac{4}{x^{2}+26}$$

Factor x^{2}+14x+33.

$$\frac{x-5}{\left(x+3\right)\left(x+11\right)}+\frac{4}{x^{2}+26}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+3\right)\left(x+11\right) and x^{2}+26 is \left(x+3\right)\left(x+11\right)\left(x^{2}+26\right). Multiply \frac{x-5}{\left(x+3\right)\left(x+11\right)} times \frac{x^{2}+26}{x^{2}+26}. Multiply \frac{4}{x^{2}+26} times \frac{\left(x+3\right)\left(x+11\right)}{\left(x+3\right)\left(x+11\right)}.

$$\frac{\left(x-5\right)\left(x^{2}+26\right)}{\left(x+3\right)\left(x+11\right)\left(x^{2}+26\right)}+\frac{4\left(x+3\right)\left(x+11\right)}{\left(x+3\right)\left(x+11\right)\left(x^{2}+26\right)}$$

Since \frac{\left(x-5\right)\left(x^{2}+26\right)}{\left(x+3\right)\left(x+11\right)\left(x^{2}+26\right)} and \frac{4\left(x+3\right)\left(x+11\right)}{\left(x+3\right)\left(x+11\right)\left(x^{2}+26\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x-5\right)\left(x^{2}+26\right)+4\left(x+3\right)\left(x+11\right)}{\left(x+3\right)\left(x+11\right)\left(x^{2}+26\right)}$$

Do the multiplications in \left(x-5\right)\left(x^{2}+26\right)+4\left(x+3\right)\left(x+11\right).

$$\frac{x^{3}+26x-5x^{2}-130+4x^{2}+44x+12x+132}{\left(x+3\right)\left(x+11\right)\left(x^{2}+26\right)}$$

Combine like terms in x^{3}+26x-5x^{2}-130+4x^{2}+44x+12x+132.

$$\frac{x^{3}+82x-x^{2}+2}{\left(x+3\right)\left(x+11\right)\left(x^{2}+26\right)}$$

Expand \left(x+3\right)\left(x+11\right)\left(x^{2}+26\right).

$$\frac{x^{3}+82x-x^{2}+2}{x^{4}+14x^{3}+59x^{2}+364x+858}$$