Solve y^3+2y/y^2-4divy^2+5y-6/y^2+4y-12

$$\frac { y ^ { 3 } + 2 y } { y ^ { 2 } - 4 } \div \frac { y ^ { 2 } + 5 y - 6 } { y ^ { 2 } + 4 y - 12 }$$

Evaluate

$$\frac{y\left(y^{2}+2\right)}{\left(y-1\right)\left(y+2\right)}$$

Short Solution Steps

Divide \frac{y^{3}+2y}{y^{2}-4} by \frac{y^{2}+5y-6}{y^{2}+4y-12} by multiplying \frac{y^{3}+2y}{y^{2}-4} by the reciprocal of \frac{y^{2}+5y-6}{y^{2}+4y-12}.

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

Factor the expressions that are not already factored.

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

Cancel out \left(y-2\right)\left(y+6\right) in both numerator and denominator.

$$\frac{y\left(y^{2}+2\right)}{\left(y-1\right)\left(y+2\right)}$$

Expand the expression.

$$\frac{y^{3}+2y}{y^{2}+y-2}$$

Expand

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

Short Solution Steps

Divide \frac{y^{3}+2y}{y^{2}-4} by \frac{y^{2}+5y-6}{y^{2}+4y-12} by multiplying \frac{y^{3}+2y}{y^{2}-4} by the reciprocal of \frac{y^{2}+5y-6}{y^{2}+4y-12}.

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

Factor the expressions that are not already factored.

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

Cancel out \left(y-2\right)\left(y+6\right) in both numerator and denominator.

$$\frac{y\left(y^{2}+2\right)}{\left(y-1\right)\left(y+2\right)}$$

Expand the expression.

$$\frac{y^{3}+2y}{y^{2}+y-2}$$