Solve 5(x+3)/(x+3)^2+9-5/2[x+3/(x+3)^2+121+x+3/(x+3)^2+25]

$$\frac { 5 ( x + 3 ) } { ( x + 3 ) ^ { 2 } + 9 } - \frac { 5 } { 2 } [ \frac { x + 3 } { ( x + 3 ) ^ { 2 } + 121 } + \frac { x + 3 } { ( x + 3 ) ^ { 2 } + 25 } ]$$

Evaluate

$$\frac{320\left(x+3\right)\left(x^{2}+6x+46\right)}{\left(x^{2}+6x+18\right)\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+3\right)^{2}+121 and \left(x+3\right)^{2}+25 is \left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right). Multiply \frac{x+3}{\left(x+3\right)^{2}+121} times \frac{\left(x+3\right)^{2}+25}{\left(x+3\right)^{2}+25}. Multiply \frac{x+3}{\left(x+3\right)^{2}+25} times \frac{\left(x+3\right)^{2}+121}{\left(x+3\right)^{2}+121}.

$$\frac{5\left(x+3\right)}{\left(x+3\right)^{2}+9}-\frac{5}{2}\left(\frac{\left(x+3\right)\left(\left(x+3\right)^{2}+25\right)}{\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)}+\frac{\left(x+3\right)\left(\left(x+3\right)^{2}+121\right)}{\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)}\right)$$

Since \frac{\left(x+3\right)\left(\left(x+3\right)^{2}+25\right)}{\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)} and \frac{\left(x+3\right)\left(\left(x+3\right)^{2}+121\right)}{\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)} have the same denominator, add them by adding their numerators.

$$\frac{5\left(x+3\right)}{\left(x+3\right)^{2}+9}-\frac{5}{2}\times \frac{\left(x+3\right)\left(\left(x+3\right)^{2}+25\right)+\left(x+3\right)\left(\left(x+3\right)^{2}+121\right)}{\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)}$$

Do the multiplications in \left(x+3\right)\left(\left(x+3\right)^{2}+25\right)+\left(x+3\right)\left(\left(x+3\right)^{2}+121\right).

$$\frac{5\left(x+3\right)}{\left(x+3\right)^{2}+9}-\frac{5}{2}\times \frac{x^{3}+6x^{2}+34x+3x^{2}+18x+102+x^{3}+6x^{2}+130x+3x^{2}+18x+390}{\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)}$$

Combine like terms in x^{3}+6x^{2}+34x+3x^{2}+18x+102+x^{3}+6x^{2}+130x+3x^{2}+18x+390.

$$\frac{5\left(x+3\right)}{\left(x+3\right)^{2}+9}-\frac{5}{2}\times \frac{2x^{3}+18x^{2}+200x+492}{\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)}$$

Multiply \frac{5}{2} times \frac{2x^{3}+18x^{2}+200x+492}{\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5\left(x+3\right)}{\left(x+3\right)^{2}+9}-\frac{5\left(2x^{3}+18x^{2}+200x+492\right)}{2\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)}$$

Factor the expressions that are not already factored in \frac{5\left(2x^{3}+18x^{2}+200x+492\right)}{2\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)}.

$$\frac{5\left(x+3\right)}{\left(x+3\right)^{2}+9}-\frac{2\times 5\left(x+3\right)\left(x^{2}+6x+82\right)}{2\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{5\left(x+3\right)}{\left(x+3\right)^{2}+9}-\frac{5\left(x+3\right)\left(x^{2}+6x+82\right)}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+3\right)^{2}+9 and \left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right) is \left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right). Multiply \frac{5\left(x+3\right)}{\left(x+3\right)^{2}+9} times \frac{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)}. Multiply \frac{5\left(x+3\right)\left(x^{2}+6x+82\right)}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)} times \frac{\left(x+3\right)^{2}+9}{\left(x+3\right)^{2}+9}.

$$\frac{5\left(x+3\right)\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right)}-\frac{5\left(x+3\right)\left(x^{2}+6x+82\right)\left(\left(x+3\right)^{2}+9\right)}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right)}$$

Since \frac{5\left(x+3\right)\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right)} and \frac{5\left(x+3\right)\left(x^{2}+6x+82\right)\left(\left(x+3\right)^{2}+9\right)}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\left(x+3\right)\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)-5\left(x+3\right)\left(x^{2}+6x+82\right)\left(\left(x+3\right)^{2}+9\right)}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right)}$$

Do the multiplications in 5\left(x+3\right)\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)-5\left(x+3\right)\left(x^{2}+6x+82\right)\left(\left(x+3\right)^{2}+9\right).

$$\frac{5x^{5}+60x^{4}+1000x^{3}+4920x^{2}+22100x+15x^{4}+180x^{3}+3000x^{2}+14760x+66300-5x^{5}-60x^{4}-680x^{3}-3000x^{2}-7380x-15x^{4}-180x^{3}-2040x^{2}-9000x-22140}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right)}$$

Combine like terms in 5x^{5}+60x^{4}+1000x^{3}+4920x^{2}+22100x+15x^{4}+180x^{3}+3000x^{2}+14760x+66300-5x^{5}-60x^{4}-680x^{3}-3000x^{2}-7380x-15x^{4}-180x^{3}-2040x^{2}-9000x-22140.

$$\frac{320x^{3}+2880x^{2}+20480x+44160}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right)}$$

Expand \left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right).

$$\frac{320x^{3}+2880x^{2}+20480x+44160}{x^{6}+18x^{5}+290x^{4}+2400x^{3}+13924x^{2}+44232x+79560}$$

Expand

$$\frac{320\left(x^{3}+9x^{2}+64x+138\right)}{\left(x^{2}+6x+18\right)\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+3\right)^{2}+121 and \left(x+3\right)^{2}+25 is \left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right). Multiply \frac{x+3}{\left(x+3\right)^{2}+121} times \frac{\left(x+3\right)^{2}+25}{\left(x+3\right)^{2}+25}. Multiply \frac{x+3}{\left(x+3\right)^{2}+25} times \frac{\left(x+3\right)^{2}+121}{\left(x+3\right)^{2}+121}.

$$\frac{5\left(x+3\right)}{\left(x+3\right)^{2}+9}-\frac{5}{2}\left(\frac{\left(x+3\right)\left(\left(x+3\right)^{2}+25\right)}{\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)}+\frac{\left(x+3\right)\left(\left(x+3\right)^{2}+121\right)}{\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)}\right)$$

Since \frac{\left(x+3\right)\left(\left(x+3\right)^{2}+25\right)}{\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)} and \frac{\left(x+3\right)\left(\left(x+3\right)^{2}+121\right)}{\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)} have the same denominator, add them by adding their numerators.

$$\frac{5\left(x+3\right)}{\left(x+3\right)^{2}+9}-\frac{5}{2}\times \frac{\left(x+3\right)\left(\left(x+3\right)^{2}+25\right)+\left(x+3\right)\left(\left(x+3\right)^{2}+121\right)}{\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)}$$

Do the multiplications in \left(x+3\right)\left(\left(x+3\right)^{2}+25\right)+\left(x+3\right)\left(\left(x+3\right)^{2}+121\right).

$$\frac{5\left(x+3\right)}{\left(x+3\right)^{2}+9}-\frac{5}{2}\times \frac{x^{3}+6x^{2}+34x+3x^{2}+18x+102+x^{3}+6x^{2}+130x+3x^{2}+18x+390}{\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)}$$

Combine like terms in x^{3}+6x^{2}+34x+3x^{2}+18x+102+x^{3}+6x^{2}+130x+3x^{2}+18x+390.

$$\frac{5\left(x+3\right)}{\left(x+3\right)^{2}+9}-\frac{5}{2}\times \frac{2x^{3}+18x^{2}+200x+492}{\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)}$$

Multiply \frac{5}{2} times \frac{2x^{3}+18x^{2}+200x+492}{\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5\left(x+3\right)}{\left(x+3\right)^{2}+9}-\frac{5\left(2x^{3}+18x^{2}+200x+492\right)}{2\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)}$$

Factor the expressions that are not already factored in \frac{5\left(2x^{3}+18x^{2}+200x+492\right)}{2\left(\left(x+3\right)^{2}+25\right)\left(\left(x+3\right)^{2}+121\right)}.

$$\frac{5\left(x+3\right)}{\left(x+3\right)^{2}+9}-\frac{2\times 5\left(x+3\right)\left(x^{2}+6x+82\right)}{2\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{5\left(x+3\right)}{\left(x+3\right)^{2}+9}-\frac{5\left(x+3\right)\left(x^{2}+6x+82\right)}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+3\right)^{2}+9 and \left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right) is \left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right). Multiply \frac{5\left(x+3\right)}{\left(x+3\right)^{2}+9} times \frac{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)}. Multiply \frac{5\left(x+3\right)\left(x^{2}+6x+82\right)}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)} times \frac{\left(x+3\right)^{2}+9}{\left(x+3\right)^{2}+9}.

$$\frac{5\left(x+3\right)\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right)}-\frac{5\left(x+3\right)\left(x^{2}+6x+82\right)\left(\left(x+3\right)^{2}+9\right)}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right)}$$

Since \frac{5\left(x+3\right)\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right)} and \frac{5\left(x+3\right)\left(x^{2}+6x+82\right)\left(\left(x+3\right)^{2}+9\right)}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\left(x+3\right)\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)-5\left(x+3\right)\left(x^{2}+6x+82\right)\left(\left(x+3\right)^{2}+9\right)}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right)}$$

Do the multiplications in 5\left(x+3\right)\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)-5\left(x+3\right)\left(x^{2}+6x+82\right)\left(\left(x+3\right)^{2}+9\right).

$$\frac{5x^{5}+60x^{4}+1000x^{3}+4920x^{2}+22100x+15x^{4}+180x^{3}+3000x^{2}+14760x+66300-5x^{5}-60x^{4}-680x^{3}-3000x^{2}-7380x-15x^{4}-180x^{3}-2040x^{2}-9000x-22140}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right)}$$

Combine like terms in 5x^{5}+60x^{4}+1000x^{3}+4920x^{2}+22100x+15x^{4}+180x^{3}+3000x^{2}+14760x+66300-5x^{5}-60x^{4}-680x^{3}-3000x^{2}-7380x-15x^{4}-180x^{3}-2040x^{2}-9000x-22140.

$$\frac{320x^{3}+2880x^{2}+20480x+44160}{\left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right)}$$

Expand \left(x^{2}+6x+34\right)\left(x^{2}+6x+130\right)\left(\left(x+3\right)^{2}+9\right).

$$\frac{320x^{3}+2880x^{2}+20480x+44160}{x^{6}+18x^{5}+290x^{4}+2400x^{3}+13924x^{2}+44232x+79560}$$