Solve (3/y^-4+4y-6/y^2-3y-4+3y/y+1)y/2y-3=

$$( \frac { 3 } { y ^ { - 4 } } + \frac { 4 y - 6 } { y ^ { 2 } - 3 y - 4 } + \frac { 3 y } { y + 1 } ) \frac { y } { 2 y - 3 } =$$

Evaluate

$$\frac{y\left(3y^{6}-9y^{5}-12y^{4}+3y^{2}-8y-6\right)}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)}$$

Short Solution Steps

Factor y^{2}-3y-4.

$$\left(\frac{3}{y^{-4}}+\frac{4y-6}{\left(y-4\right)\left(y+1\right)}+\frac{3y}{y+1}\right)\times \frac{y}{2y-3}$$

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

$$\left(\frac{3}{y^{-4}}+\frac{4y-6}{\left(y-4\right)\left(y+1\right)}+\frac{3y\left(y-4\right)}{\left(y-4\right)\left(y+1\right)}\right)\times \frac{y}{2y-3}$$

Since \frac{4y-6}{\left(y-4\right)\left(y+1\right)} and \frac{3y\left(y-4\right)}{\left(y-4\right)\left(y+1\right)} have the same denominator, add them by adding their numerators.

$$\left(\frac{3}{y^{-4}}+\frac{4y-6+3y\left(y-4\right)}{\left(y-4\right)\left(y+1\right)}\right)\times \frac{y}{2y-3}$$

Do the multiplications in 4y-6+3y\left(y-4\right).

$$\left(\frac{3}{y^{-4}}+\frac{4y-6+3y^{2}-12y}{\left(y-4\right)\left(y+1\right)}\right)\times \frac{y}{2y-3}$$

Combine like terms in 4y-6+3y^{2}-12y.

$$\left(\frac{3}{y^{-4}}+\frac{-8y-6+3y^{2}}{\left(y-4\right)\left(y+1\right)}\right)\times \frac{y}{2y-3}$$

Use the distributive property to multiply \frac{3}{y^{-4}}+\frac{-8y-6+3y^{2}}{\left(y-4\right)\left(y+1\right)} by \frac{y}{2y-3}.

$$\frac{3}{y^{-4}}\times \frac{y}{2y-3}+\frac{-8y-6+3y^{2}}{\left(y-4\right)\left(y+1\right)}\times \frac{y}{2y-3}$$

Multiply \frac{3}{y^{-4}} times \frac{y}{2y-3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3y}{y^{-4}\left(2y-3\right)}+\frac{-8y-6+3y^{2}}{\left(y-4\right)\left(y+1\right)}\times \frac{y}{2y-3}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{3y^{5}}{2y-3}+\frac{-8y-6+3y^{2}}{\left(y-4\right)\left(y+1\right)}\times \frac{y}{2y-3}$$

Use the distributive property to multiply y-4 by y+1 and combine like terms.

$$\frac{3y^{5}}{2y-3}+\frac{-8y-6+3y^{2}}{y^{2}-3y-4}\times \frac{y}{2y-3}$$

Multiply \frac{-8y-6+3y^{2}}{y^{2}-3y-4} times \frac{y}{2y-3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3y^{5}}{2y-3}+\frac{\left(-8y-6+3y^{2}\right)y}{\left(y^{2}-3y-4\right)\left(2y-3\right)}$$

Factor \left(y^{2}-3y-4\right)\left(2y-3\right).

$$\frac{3y^{5}}{2y-3}+\frac{\left(-8y-6+3y^{2}\right)y}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2y-3 and \left(y-4\right)\left(2y-3\right)\left(y+1\right) is \left(y-4\right)\left(2y-3\right)\left(y+1\right). Multiply \frac{3y^{5}}{2y-3} times \frac{\left(y-4\right)\left(y+1\right)}{\left(y-4\right)\left(y+1\right)}.

$$\frac{3y^{5}\left(y-4\right)\left(y+1\right)}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)}+\frac{\left(-8y-6+3y^{2}\right)y}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)}$$

Since \frac{3y^{5}\left(y-4\right)\left(y+1\right)}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)} and \frac{\left(-8y-6+3y^{2}\right)y}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{3y^{5}\left(y-4\right)\left(y+1\right)+\left(-8y-6+3y^{2}\right)y}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)}$$

Do the multiplications in 3y^{5}\left(y-4\right)\left(y+1\right)+\left(-8y-6+3y^{2}\right)y.

$$\frac{3y^{7}+3y^{6}-12y^{6}-12y^{5}-8y^{2}-6y+3y^{3}}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)}$$

Combine like terms in 3y^{7}+3y^{6}-12y^{6}-12y^{5}-8y^{2}-6y+3y^{3}.

$$\frac{3y^{7}-6y-9y^{6}-12y^{5}-8y^{2}+3y^{3}}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)}$$

Expand \left(y-4\right)\left(2y-3\right)\left(y+1\right).

$$\frac{3y^{7}-6y-9y^{6}-12y^{5}-8y^{2}+3y^{3}}{2y^{3}-9y^{2}+y+12}$$

Expand

$$\frac{3y^{7}-9y^{6}-12y^{5}+3y^{3}-8y^{2}-6y}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)}$$

Short Solution Steps

Factor y^{2}-3y-4.

$$\left(\frac{3}{y^{-4}}+\frac{4y-6}{\left(y-4\right)\left(y+1\right)}+\frac{3y}{y+1}\right)\times \frac{y}{2y-3}$$

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

$$\left(\frac{3}{y^{-4}}+\frac{4y-6}{\left(y-4\right)\left(y+1\right)}+\frac{3y\left(y-4\right)}{\left(y-4\right)\left(y+1\right)}\right)\times \frac{y}{2y-3}$$

Since \frac{4y-6}{\left(y-4\right)\left(y+1\right)} and \frac{3y\left(y-4\right)}{\left(y-4\right)\left(y+1\right)} have the same denominator, add them by adding their numerators.

$$\left(\frac{3}{y^{-4}}+\frac{4y-6+3y\left(y-4\right)}{\left(y-4\right)\left(y+1\right)}\right)\times \frac{y}{2y-3}$$

Do the multiplications in 4y-6+3y\left(y-4\right).

$$\left(\frac{3}{y^{-4}}+\frac{4y-6+3y^{2}-12y}{\left(y-4\right)\left(y+1\right)}\right)\times \frac{y}{2y-3}$$

Combine like terms in 4y-6+3y^{2}-12y.

$$\left(\frac{3}{y^{-4}}+\frac{-8y-6+3y^{2}}{\left(y-4\right)\left(y+1\right)}\right)\times \frac{y}{2y-3}$$

Use the distributive property to multiply \frac{3}{y^{-4}}+\frac{-8y-6+3y^{2}}{\left(y-4\right)\left(y+1\right)} by \frac{y}{2y-3}.

$$\frac{3}{y^{-4}}\times \frac{y}{2y-3}+\frac{-8y-6+3y^{2}}{\left(y-4\right)\left(y+1\right)}\times \frac{y}{2y-3}$$

Multiply \frac{3}{y^{-4}} times \frac{y}{2y-3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3y}{y^{-4}\left(2y-3\right)}+\frac{-8y-6+3y^{2}}{\left(y-4\right)\left(y+1\right)}\times \frac{y}{2y-3}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{3y^{5}}{2y-3}+\frac{-8y-6+3y^{2}}{\left(y-4\right)\left(y+1\right)}\times \frac{y}{2y-3}$$

Use the distributive property to multiply y-4 by y+1 and combine like terms.

$$\frac{3y^{5}}{2y-3}+\frac{-8y-6+3y^{2}}{y^{2}-3y-4}\times \frac{y}{2y-3}$$

Multiply \frac{-8y-6+3y^{2}}{y^{2}-3y-4} times \frac{y}{2y-3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3y^{5}}{2y-3}+\frac{\left(-8y-6+3y^{2}\right)y}{\left(y^{2}-3y-4\right)\left(2y-3\right)}$$

Factor \left(y^{2}-3y-4\right)\left(2y-3\right).

$$\frac{3y^{5}}{2y-3}+\frac{\left(-8y-6+3y^{2}\right)y}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2y-3 and \left(y-4\right)\left(2y-3\right)\left(y+1\right) is \left(y-4\right)\left(2y-3\right)\left(y+1\right). Multiply \frac{3y^{5}}{2y-3} times \frac{\left(y-4\right)\left(y+1\right)}{\left(y-4\right)\left(y+1\right)}.

$$\frac{3y^{5}\left(y-4\right)\left(y+1\right)}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)}+\frac{\left(-8y-6+3y^{2}\right)y}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)}$$

Since \frac{3y^{5}\left(y-4\right)\left(y+1\right)}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)} and \frac{\left(-8y-6+3y^{2}\right)y}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{3y^{5}\left(y-4\right)\left(y+1\right)+\left(-8y-6+3y^{2}\right)y}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)}$$

Do the multiplications in 3y^{5}\left(y-4\right)\left(y+1\right)+\left(-8y-6+3y^{2}\right)y.

$$\frac{3y^{7}+3y^{6}-12y^{6}-12y^{5}-8y^{2}-6y+3y^{3}}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)}$$

Combine like terms in 3y^{7}+3y^{6}-12y^{6}-12y^{5}-8y^{2}-6y+3y^{3}.

$$\frac{3y^{7}-6y-9y^{6}-12y^{5}-8y^{2}+3y^{3}}{\left(y-4\right)\left(2y-3\right)\left(y+1\right)}$$

Expand \left(y-4\right)\left(2y-3\right)\left(y+1\right).

$$\frac{3y^{7}-6y-9y^{6}-12y^{5}-8y^{2}+3y^{3}}{2y^{3}-9y^{2}+y+12}$$