Solve [1.5-(x^{4}-frac{x^{4}+1}{x^{2}+1})*frac{(x^{2}+1)(x-4)}{x^7+6x^6-x-6}]:x^2+29x+78/3x^2+12x-36=

$$[ 1.5 - ( x ^ { 4 } - \frac { x ^ { 4 } + 1 } { x ^ { 2 } + 1 } ) \cdot \frac { ( x ^ { 2 } + 1 ) ( x - 4 ) } { x ^ { 7 } + 6 x ^ { 6 } - x - 6 } ] : \frac { x ^ { 2 } + 29 x + 78 } { 3 x ^ { 2 } + 12 x - 36 } =$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{4} times \frac{x^{2}+1}{x^{2}+1}.

$$\frac{1.5-\left(\frac{x^{4}\left(x^{2}+1\right)}{x^{2}+1}-\frac{x^{4}+1}{x^{2}+1}\right)\times \frac{\left(x^{2}+1\right)\left(x-4\right)}{x^{7}+6x^{6}-x-6}}{\frac{x^{2}+29x+78}{3x^{2}+12x-36}}$$

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

$$\frac{1.5-\frac{x^{4}\left(x^{2}+1\right)-\left(x^{4}+1\right)}{x^{2}+1}\times \frac{\left(x^{2}+1\right)\left(x-4\right)}{x^{7}+6x^{6}-x-6}}{\frac{x^{2}+29x+78}{3x^{2}+12x-36}}$$

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

$$\frac{1.5-\frac{x^{6}+x^{4}-x^{4}-1}{x^{2}+1}\times \frac{\left(x^{2}+1\right)\left(x-4\right)}{x^{7}+6x^{6}-x-6}}{\frac{x^{2}+29x+78}{3x^{2}+12x-36}}$$

Combine like terms in x^{6}+x^{4}-x^{4}-1.

$$\frac{1.5-\frac{x^{6}-1}{x^{2}+1}\times \frac{\left(x^{2}+1\right)\left(x-4\right)}{x^{7}+6x^{6}-x-6}}{\frac{x^{2}+29x+78}{3x^{2}+12x-36}}$$

Multiply \frac{x^{6}-1}{x^{2}+1} times \frac{\left(x^{2}+1\right)\left(x-4\right)}{x^{7}+6x^{6}-x-6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1.5-\frac{\left(x^{6}-1\right)\left(x^{2}+1\right)\left(x-4\right)}{\left(x^{2}+1\right)\left(x^{7}+6x^{6}-x-6\right)}}{\frac{x^{2}+29x+78}{3x^{2}+12x-36}}$$

Cancel out x^{2}+1 in both numerator and denominator.

$$\frac{1.5-\frac{\left(x-4\right)\left(x^{6}-1\right)}{x^{7}+6x^{6}-x-6}}{\frac{x^{2}+29x+78}{3x^{2}+12x-36}}$$

Factor the expressions that are not already factored in \frac{\left(x-4\right)\left(x^{6}-1\right)}{x^{7}+6x^{6}-x-6}.

$$\frac{1.5-\frac{\left(x-4\right)\left(x-1\right)\left(x+1\right)\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}{\left(x-1\right)\left(x+1\right)\left(x+6\right)\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}}{\frac{x^{2}+29x+78}{3x^{2}+12x-36}}$$

Cancel out \left(x-1\right)\left(x+1\right)\left(x^{2}+x+1\right)\left(x^{2}-x+1\right) in both numerator and denominator.

$$\frac{1.5-\frac{x-4}{x+6}}{\frac{x^{2}+29x+78}{3x^{2}+12x-36}}$$

Divide 1.5-\frac{x-4}{x+6} by \frac{x^{2}+29x+78}{3x^{2}+12x-36} by multiplying 1.5-\frac{x-4}{x+6} by the reciprocal of \frac{x^{2}+29x+78}{3x^{2}+12x-36}.

$$\frac{\left(1.5-\frac{x-4}{x+6}\right)\left(3x^{2}+12x-36\right)}{x^{2}+29x+78}$$

Use the distributive property to multiply 1.5-\frac{x-4}{x+6} by 3x^{2}+12x-36.

$$\frac{4.5x^{2}+18x-54-3\times \frac{x-4}{x+6}x^{2}-12\times \frac{x-4}{x+6}x+36\times \frac{x-4}{x+6}}{x^{2}+29x+78}$$

Express -3\times \frac{x-4}{x+6} as a single fraction.

$$\frac{4.5x^{2}+18x-54+\frac{-3\left(x-4\right)}{x+6}x^{2}-12\times \frac{x-4}{x+6}x+36\times \frac{x-4}{x+6}}{x^{2}+29x+78}$$

Express \frac{-3\left(x-4\right)}{x+6}x^{2} as a single fraction.

$$\frac{4.5x^{2}+18x-54+\frac{-3\left(x-4\right)x^{2}}{x+6}-12\times \frac{x-4}{x+6}x+36\times \frac{x-4}{x+6}}{x^{2}+29x+78}$$

Express -12\times \frac{x-4}{x+6} as a single fraction.

$$\frac{4.5x^{2}+18x-54+\frac{-3\left(x-4\right)x^{2}}{x+6}+\frac{-12\left(x-4\right)}{x+6}x+36\times \frac{x-4}{x+6}}{x^{2}+29x+78}$$

Express \frac{-12\left(x-4\right)}{x+6}x as a single fraction.

$$\frac{4.5x^{2}+18x-54+\frac{-3\left(x-4\right)x^{2}}{x+6}+\frac{-12\left(x-4\right)x}{x+6}+36\times \frac{x-4}{x+6}}{x^{2}+29x+78}$$

Express 36\times \frac{x-4}{x+6} as a single fraction.

$$\frac{4.5x^{2}+18x-54+\frac{-3\left(x-4\right)x^{2}}{x+6}+\frac{-12\left(x-4\right)x}{x+6}+\frac{36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 18x-54 times \frac{x+6}{x+6}.

$$\frac{4.5x^{2}+\frac{\left(18x-54\right)\left(x+6\right)}{x+6}+\frac{-3\left(x-4\right)x^{2}}{x+6}+\frac{-12\left(x-4\right)x}{x+6}+\frac{36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

Since \frac{\left(18x-54\right)\left(x+6\right)}{x+6} and \frac{-3\left(x-4\right)x^{2}}{x+6} have the same denominator, add them by adding their numerators.

$$\frac{4.5x^{2}+\frac{\left(18x-54\right)\left(x+6\right)-3\left(x-4\right)x^{2}}{x+6}+\frac{-12\left(x-4\right)x}{x+6}+\frac{36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

Do the multiplications in \left(18x-54\right)\left(x+6\right)-3\left(x-4\right)x^{2}.

$$\frac{4.5x^{2}+\frac{18x^{2}+108x-54x-324-3x^{3}+12x^{2}}{x+6}+\frac{-12\left(x-4\right)x}{x+6}+\frac{36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

Combine like terms in 18x^{2}+108x-54x-324-3x^{3}+12x^{2}.

$$\frac{4.5x^{2}+\frac{30x^{2}+54x-324-3x^{3}}{x+6}+\frac{-12\left(x-4\right)x}{x+6}+\frac{36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

Since \frac{30x^{2}+54x-324-3x^{3}}{x+6} and \frac{-12\left(x-4\right)x}{x+6} have the same denominator, add them by adding their numerators.

$$\frac{4.5x^{2}+\frac{30x^{2}+54x-324-3x^{3}-12\left(x-4\right)x}{x+6}+\frac{36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

Do the multiplications in 30x^{2}+54x-324-3x^{3}-12\left(x-4\right)x.

$$\frac{4.5x^{2}+\frac{30x^{2}+54x-324-3x^{3}-12x^{2}+48x}{x+6}+\frac{36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

Combine like terms in 30x^{2}+54x-324-3x^{3}-12x^{2}+48x.

$$\frac{4.5x^{2}+\frac{18x^{2}+102x-324-3x^{3}}{x+6}+\frac{36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

Since \frac{18x^{2}+102x-324-3x^{3}}{x+6} and \frac{36\left(x-4\right)}{x+6} have the same denominator, add them by adding their numerators.

$$\frac{4.5x^{2}+\frac{18x^{2}+102x-324-3x^{3}+36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

Do the multiplications in 18x^{2}+102x-324-3x^{3}+36\left(x-4\right).

$$\frac{4.5x^{2}+\frac{18x^{2}+102x-324-3x^{3}+36x-144}{x+6}}{x^{2}+29x+78}$$

Combine like terms in 18x^{2}+102x-324-3x^{3}+36x-144.

$$\frac{4.5x^{2}+\frac{18x^{2}+138x-468-3x^{3}}{x+6}}{x^{2}+29x+78}$$

Factor the expressions that are not already factored in \frac{18x^{2}+138x-468-3x^{3}}{x+6}.

$$\frac{4.5x^{2}+\frac{3\left(x+6\right)\left(-x^{2}+12x-26\right)}{x+6}}{x^{2}+29x+78}$$

Cancel out x+6 in both numerator and denominator.

$$\frac{4.5x^{2}+3\left(-x^{2}+12x-26\right)}{x^{2}+29x+78}$$

Expand the expression.

$$\frac{4.5x^{2}-3x^{2}+36x-78}{x^{2}+29x+78}$$

Combine 4.5x^{2} and -3x^{2} to get 1.5x^{2}.

$$\frac{1.5x^{2}+36x-78}{x^{2}+29x+78}$$

Factor the expressions that are not already factored.

$$\frac{3\left(\frac{1}{2}x-1\right)\left(x+26\right)}{\left(x+3\right)\left(x+26\right)}$$

Cancel out x+26 in both numerator and denominator.

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

Expand the expression.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{4} times \frac{x^{2}+1}{x^{2}+1}.

$$\frac{1.5-\left(\frac{x^{4}\left(x^{2}+1\right)}{x^{2}+1}-\frac{x^{4}+1}{x^{2}+1}\right)\times \frac{\left(x^{2}+1\right)\left(x-4\right)}{x^{7}+6x^{6}-x-6}}{\frac{x^{2}+29x+78}{3x^{2}+12x-36}}$$

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

$$\frac{1.5-\frac{x^{4}\left(x^{2}+1\right)-\left(x^{4}+1\right)}{x^{2}+1}\times \frac{\left(x^{2}+1\right)\left(x-4\right)}{x^{7}+6x^{6}-x-6}}{\frac{x^{2}+29x+78}{3x^{2}+12x-36}}$$

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

$$\frac{1.5-\frac{x^{6}+x^{4}-x^{4}-1}{x^{2}+1}\times \frac{\left(x^{2}+1\right)\left(x-4\right)}{x^{7}+6x^{6}-x-6}}{\frac{x^{2}+29x+78}{3x^{2}+12x-36}}$$

Combine like terms in x^{6}+x^{4}-x^{4}-1.

$$\frac{1.5-\frac{x^{6}-1}{x^{2}+1}\times \frac{\left(x^{2}+1\right)\left(x-4\right)}{x^{7}+6x^{6}-x-6}}{\frac{x^{2}+29x+78}{3x^{2}+12x-36}}$$

Multiply \frac{x^{6}-1}{x^{2}+1} times \frac{\left(x^{2}+1\right)\left(x-4\right)}{x^{7}+6x^{6}-x-6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1.5-\frac{\left(x^{6}-1\right)\left(x^{2}+1\right)\left(x-4\right)}{\left(x^{2}+1\right)\left(x^{7}+6x^{6}-x-6\right)}}{\frac{x^{2}+29x+78}{3x^{2}+12x-36}}$$

Cancel out x^{2}+1 in both numerator and denominator.

$$\frac{1.5-\frac{\left(x-4\right)\left(x^{6}-1\right)}{x^{7}+6x^{6}-x-6}}{\frac{x^{2}+29x+78}{3x^{2}+12x-36}}$$

Factor the expressions that are not already factored in \frac{\left(x-4\right)\left(x^{6}-1\right)}{x^{7}+6x^{6}-x-6}.

$$\frac{1.5-\frac{\left(x-4\right)\left(x-1\right)\left(x+1\right)\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}{\left(x-1\right)\left(x+1\right)\left(x+6\right)\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)}}{\frac{x^{2}+29x+78}{3x^{2}+12x-36}}$$

Cancel out \left(x-1\right)\left(x+1\right)\left(x^{2}+x+1\right)\left(x^{2}-x+1\right) in both numerator and denominator.

$$\frac{1.5-\frac{x-4}{x+6}}{\frac{x^{2}+29x+78}{3x^{2}+12x-36}}$$

Divide 1.5-\frac{x-4}{x+6} by \frac{x^{2}+29x+78}{3x^{2}+12x-36} by multiplying 1.5-\frac{x-4}{x+6} by the reciprocal of \frac{x^{2}+29x+78}{3x^{2}+12x-36}.

$$\frac{\left(1.5-\frac{x-4}{x+6}\right)\left(3x^{2}+12x-36\right)}{x^{2}+29x+78}$$

Use the distributive property to multiply 1.5-\frac{x-4}{x+6} by 3x^{2}+12x-36.

$$\frac{4.5x^{2}+18x-54-3\times \frac{x-4}{x+6}x^{2}-12\times \frac{x-4}{x+6}x+36\times \frac{x-4}{x+6}}{x^{2}+29x+78}$$

Express -3\times \frac{x-4}{x+6} as a single fraction.

$$\frac{4.5x^{2}+18x-54+\frac{-3\left(x-4\right)}{x+6}x^{2}-12\times \frac{x-4}{x+6}x+36\times \frac{x-4}{x+6}}{x^{2}+29x+78}$$

Express \frac{-3\left(x-4\right)}{x+6}x^{2} as a single fraction.

$$\frac{4.5x^{2}+18x-54+\frac{-3\left(x-4\right)x^{2}}{x+6}-12\times \frac{x-4}{x+6}x+36\times \frac{x-4}{x+6}}{x^{2}+29x+78}$$

Express -12\times \frac{x-4}{x+6} as a single fraction.

$$\frac{4.5x^{2}+18x-54+\frac{-3\left(x-4\right)x^{2}}{x+6}+\frac{-12\left(x-4\right)}{x+6}x+36\times \frac{x-4}{x+6}}{x^{2}+29x+78}$$

Express \frac{-12\left(x-4\right)}{x+6}x as a single fraction.

$$\frac{4.5x^{2}+18x-54+\frac{-3\left(x-4\right)x^{2}}{x+6}+\frac{-12\left(x-4\right)x}{x+6}+36\times \frac{x-4}{x+6}}{x^{2}+29x+78}$$

Express 36\times \frac{x-4}{x+6} as a single fraction.

$$\frac{4.5x^{2}+18x-54+\frac{-3\left(x-4\right)x^{2}}{x+6}+\frac{-12\left(x-4\right)x}{x+6}+\frac{36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 18x-54 times \frac{x+6}{x+6}.

$$\frac{4.5x^{2}+\frac{\left(18x-54\right)\left(x+6\right)}{x+6}+\frac{-3\left(x-4\right)x^{2}}{x+6}+\frac{-12\left(x-4\right)x}{x+6}+\frac{36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

Since \frac{\left(18x-54\right)\left(x+6\right)}{x+6} and \frac{-3\left(x-4\right)x^{2}}{x+6} have the same denominator, add them by adding their numerators.

$$\frac{4.5x^{2}+\frac{\left(18x-54\right)\left(x+6\right)-3\left(x-4\right)x^{2}}{x+6}+\frac{-12\left(x-4\right)x}{x+6}+\frac{36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

Do the multiplications in \left(18x-54\right)\left(x+6\right)-3\left(x-4\right)x^{2}.

$$\frac{4.5x^{2}+\frac{18x^{2}+108x-54x-324-3x^{3}+12x^{2}}{x+6}+\frac{-12\left(x-4\right)x}{x+6}+\frac{36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

Combine like terms in 18x^{2}+108x-54x-324-3x^{3}+12x^{2}.

$$\frac{4.5x^{2}+\frac{30x^{2}+54x-324-3x^{3}}{x+6}+\frac{-12\left(x-4\right)x}{x+6}+\frac{36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

Since \frac{30x^{2}+54x-324-3x^{3}}{x+6} and \frac{-12\left(x-4\right)x}{x+6} have the same denominator, add them by adding their numerators.

$$\frac{4.5x^{2}+\frac{30x^{2}+54x-324-3x^{3}-12\left(x-4\right)x}{x+6}+\frac{36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

Do the multiplications in 30x^{2}+54x-324-3x^{3}-12\left(x-4\right)x.

$$\frac{4.5x^{2}+\frac{30x^{2}+54x-324-3x^{3}-12x^{2}+48x}{x+6}+\frac{36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

Combine like terms in 30x^{2}+54x-324-3x^{3}-12x^{2}+48x.

$$\frac{4.5x^{2}+\frac{18x^{2}+102x-324-3x^{3}}{x+6}+\frac{36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

Since \frac{18x^{2}+102x-324-3x^{3}}{x+6} and \frac{36\left(x-4\right)}{x+6} have the same denominator, add them by adding their numerators.

$$\frac{4.5x^{2}+\frac{18x^{2}+102x-324-3x^{3}+36\left(x-4\right)}{x+6}}{x^{2}+29x+78}$$

Do the multiplications in 18x^{2}+102x-324-3x^{3}+36\left(x-4\right).

$$\frac{4.5x^{2}+\frac{18x^{2}+102x-324-3x^{3}+36x-144}{x+6}}{x^{2}+29x+78}$$

Combine like terms in 18x^{2}+102x-324-3x^{3}+36x-144.

$$\frac{4.5x^{2}+\frac{18x^{2}+138x-468-3x^{3}}{x+6}}{x^{2}+29x+78}$$

Factor the expressions that are not already factored in \frac{18x^{2}+138x-468-3x^{3}}{x+6}.

$$\frac{4.5x^{2}+\frac{3\left(x+6\right)\left(-x^{2}+12x-26\right)}{x+6}}{x^{2}+29x+78}$$

Cancel out x+6 in both numerator and denominator.

$$\frac{4.5x^{2}+3\left(-x^{2}+12x-26\right)}{x^{2}+29x+78}$$

Expand the expression.

$$\frac{4.5x^{2}-3x^{2}+36x-78}{x^{2}+29x+78}$$

Combine 4.5x^{2} and -3x^{2} to get 1.5x^{2}.

$$\frac{1.5x^{2}+36x-78}{x^{2}+29x+78}$$

Factor the expressions that are not already factored.

$$\frac{3\left(\frac{1}{2}x-1\right)\left(x+26\right)}{\left(x+3\right)\left(x+26\right)}$$

Cancel out x+26 in both numerator and denominator.

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

Expand the expression.

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