Solve frac{y(x+y^{2}+y)+x}{y+1}divfrac{4y^{2}-3yx-7x^2}{y^2-x^2}*4y-7x/y^2+x

$$\frac { y ( x + y ^ { 2 } + y ) + x } { y + 1 } \div \frac { 4 y ^ { 2 } - 3 y x - 7 x ^ { 2 } } { y ^ { 2 } - x ^ { 2 } } \times \frac { 4 y - 7 x } { y ^ { 2 } + x }$$

Evaluate

$$y-x$$

Short Solution Steps

Divide \frac{y\left(x+y^{2}+y\right)+x}{y+1} by \frac{4y^{2}-3yx-7x^{2}}{y^{2}-x^{2}} by multiplying \frac{y\left(x+y^{2}+y\right)+x}{y+1} by the reciprocal of \frac{4y^{2}-3yx-7x^{2}}{y^{2}-x^{2}}.

$$\frac{\left(y\left(x+y^{2}+y\right)+x\right)\left(y^{2}-x^{2}\right)}{\left(y+1\right)\left(4y^{2}-3yx-7x^{2}\right)}\times \frac{4y-7x}{y^{2}+x}$$

Factor the expressions that are not already factored in \frac{\left(y\left(x+y^{2}+y\right)+x\right)\left(y^{2}-x^{2}\right)}{\left(y+1\right)\left(4y^{2}-3yx-7x^{2}\right)}.

$$\frac{\left(y+1\right)\left(x+y\right)\left(-x+y\right)\left(x+y^{2}\right)}{\left(y+1\right)\left(x+y\right)\left(-7x+4y\right)}\times \frac{4y-7x}{y^{2}+x}$$

Cancel out \left(y+1\right)\left(x+y\right) in both numerator and denominator.

$$\frac{\left(-x+y\right)\left(x+y^{2}\right)}{-7x+4y}\times \frac{4y-7x}{y^{2}+x}$$

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

$$\frac{\left(-x+y\right)\left(x+y^{2}\right)\left(4y-7x\right)}{\left(-7x+4y\right)\left(y^{2}+x\right)}$$

Cancel out \left(-7x+4y\right)\left(x+y^{2}\right) in both numerator and denominator.

$$-x+y$$

Expand

$$y-x$$

Short Solution Steps

Divide \frac{y\left(x+y^{2}+y\right)+x}{y+1} by \frac{4y^{2}-3yx-7x^{2}}{y^{2}-x^{2}} by multiplying \frac{y\left(x+y^{2}+y\right)+x}{y+1} by the reciprocal of \frac{4y^{2}-3yx-7x^{2}}{y^{2}-x^{2}}.

$$\frac{\left(y\left(x+y^{2}+y\right)+x\right)\left(y^{2}-x^{2}\right)}{\left(y+1\right)\left(4y^{2}-3yx-7x^{2}\right)}\times \frac{4y-7x}{y^{2}+x}$$

Factor the expressions that are not already factored in \frac{\left(y\left(x+y^{2}+y\right)+x\right)\left(y^{2}-x^{2}\right)}{\left(y+1\right)\left(4y^{2}-3yx-7x^{2}\right)}.

$$\frac{\left(y+1\right)\left(x+y\right)\left(-x+y\right)\left(x+y^{2}\right)}{\left(y+1\right)\left(x+y\right)\left(-7x+4y\right)}\times \frac{4y-7x}{y^{2}+x}$$

Cancel out \left(y+1\right)\left(x+y\right) in both numerator and denominator.

$$\frac{\left(-x+y\right)\left(x+y^{2}\right)}{-7x+4y}\times \frac{4y-7x}{y^{2}+x}$$

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

$$\frac{\left(-x+y\right)\left(x+y^{2}\right)\left(4y-7x\right)}{\left(-7x+4y\right)\left(y^{2}+x\right)}$$

Cancel out \left(-7x+4y\right)\left(x+y^{2}\right) in both numerator and denominator.

$$-x+y$$