Evaluate
-\frac{2\left(y^{2}+1\right)}{3\left(y^{2}-1\right)}
Expand
-\frac{2\left(y^{2}+1\right)}{3\left(y^{2}-1\right)}
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\left(\frac{x\left(y-1\right)}{\left(y-1\right)\left(y+1\right)}-\frac{x\left(y+1\right)}{\left(y-1\right)\left(y+1\right)}\right)\times \frac{x+xy^{2}}{3x^{2}}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+1 and y-1 is \left(y-1\right)\left(y+1\right). Multiply \frac{x}{y+1} times \frac{y-1}{y-1}. Multiply \frac{x}{y-1} times \frac{y+1}{y+1}.
\frac{x\left(y-1\right)-x\left(y+1\right)}{\left(y-1\right)\left(y+1\right)}\times \frac{x+xy^{2}}{3x^{2}}
Since \frac{x\left(y-1\right)}{\left(y-1\right)\left(y+1\right)} and \frac{x\left(y+1\right)}{\left(y-1\right)\left(y+1\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{xy-x-xy-x}{\left(y-1\right)\left(y+1\right)}\times \frac{x+xy^{2}}{3x^{2}}
Do the multiplications in x\left(y-1\right)-x\left(y+1\right).
\frac{-2x}{\left(y-1\right)\left(y+1\right)}\times \frac{x+xy^{2}}{3x^{2}}
Combine like terms in xy-x-xy-x.
\frac{-2x}{\left(y-1\right)\left(y+1\right)}\times \frac{x\left(y^{2}+1\right)}{3x^{2}}
Factor the expressions that are not already factored in \frac{x+xy^{2}}{3x^{2}}.
\frac{-2x}{\left(y-1\right)\left(y+1\right)}\times \frac{y^{2}+1}{3x}
Cancel out x in both numerator and denominator.
\frac{-2x\left(y^{2}+1\right)}{\left(y-1\right)\left(y+1\right)\times 3x}
Multiply \frac{-2x}{\left(y-1\right)\left(y+1\right)} times \frac{y^{2}+1}{3x} by multiplying numerator times numerator and denominator times denominator.
\frac{-2\left(y^{2}+1\right)}{3\left(y-1\right)\left(y+1\right)}
Cancel out x in both numerator and denominator.
\frac{-2y^{2}-2}{3\left(y-1\right)\left(y+1\right)}
Use the distributive property to multiply -2 by y^{2}+1.
\frac{-2y^{2}-2}{\left(3y-3\right)\left(y+1\right)}
Use the distributive property to multiply 3 by y-1.
\frac{-2y^{2}-2}{3y^{2}-3}
Use the distributive property to multiply 3y-3 by y+1 and combine like terms.
\left(\frac{x\left(y-1\right)}{\left(y-1\right)\left(y+1\right)}-\frac{x\left(y+1\right)}{\left(y-1\right)\left(y+1\right)}\right)\times \frac{x+xy^{2}}{3x^{2}}
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+1 and y-1 is \left(y-1\right)\left(y+1\right). Multiply \frac{x}{y+1} times \frac{y-1}{y-1}. Multiply \frac{x}{y-1} times \frac{y+1}{y+1}.
\frac{x\left(y-1\right)-x\left(y+1\right)}{\left(y-1\right)\left(y+1\right)}\times \frac{x+xy^{2}}{3x^{2}}
Since \frac{x\left(y-1\right)}{\left(y-1\right)\left(y+1\right)} and \frac{x\left(y+1\right)}{\left(y-1\right)\left(y+1\right)} have the same denominator, subtract them by subtracting their numerators.
\frac{xy-x-xy-x}{\left(y-1\right)\left(y+1\right)}\times \frac{x+xy^{2}}{3x^{2}}
Do the multiplications in x\left(y-1\right)-x\left(y+1\right).
\frac{-2x}{\left(y-1\right)\left(y+1\right)}\times \frac{x+xy^{2}}{3x^{2}}
Combine like terms in xy-x-xy-x.
\frac{-2x}{\left(y-1\right)\left(y+1\right)}\times \frac{x\left(y^{2}+1\right)}{3x^{2}}
Factor the expressions that are not already factored in \frac{x+xy^{2}}{3x^{2}}.
\frac{-2x}{\left(y-1\right)\left(y+1\right)}\times \frac{y^{2}+1}{3x}
Cancel out x in both numerator and denominator.
\frac{-2x\left(y^{2}+1\right)}{\left(y-1\right)\left(y+1\right)\times 3x}
Multiply \frac{-2x}{\left(y-1\right)\left(y+1\right)} times \frac{y^{2}+1}{3x} by multiplying numerator times numerator and denominator times denominator.
\frac{-2\left(y^{2}+1\right)}{3\left(y-1\right)\left(y+1\right)}
Cancel out x in both numerator and denominator.
\frac{-2y^{2}-2}{3\left(y-1\right)\left(y+1\right)}
Use the distributive property to multiply -2 by y^{2}+1.
\frac{-2y^{2}-2}{\left(3y-3\right)\left(y+1\right)}
Use the distributive property to multiply 3 by y-1.
\frac{-2y^{2}-2}{3y^{2}-3}
Use the distributive property to multiply 3y-3 by y+1 and combine like terms.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}