Solve (-4)*frac{12y_{0}-(27+4y_{0})y_{0}}{27-4(27+4y_0^2)}+y_{0}

$$( - 4 ) \times \frac { 12 y _ { 0 } - ( 27 + 4 y _ { 0 } ) y _ { 0 } } { 27 - 4 ( 27 + 4 y _ { 0 } ^ { 2 } ) } + y _ { 0 }$$

Evaluate

$$-\frac{y_{0}\left(-16y_{0}^{2}+16y_{0}-21\right)}{16y_{0}^{2}+81}$$

Short Solution Steps

Use the distributive property to multiply 27+4y_{0} by y_{0}.

$$-4\times \frac{12y_{0}-\left(27y_{0}+4y_{0}^{2}\right)}{27-4\left(27+4y_{0}^{2}\right)}+y_{0}$$

To find the opposite of 27y_{0}+4y_{0}^{2}, find the opposite of each term.

$$-4\times \frac{12y_{0}-27y_{0}-4y_{0}^{2}}{27-4\left(27+4y_{0}^{2}\right)}+y_{0}$$

Combine 12y_{0} and -27y_{0} to get -15y_{0}.

$$-4\times \frac{-15y_{0}-4y_{0}^{2}}{27-4\left(27+4y_{0}^{2}\right)}+y_{0}$$

Use the distributive property to multiply -4 by 27+4y_{0}^{2}.

$$-4\times \frac{-15y_{0}-4y_{0}^{2}}{27-108-16y_{0}^{2}}+y_{0}$$

Subtract 108 from 27 to get -81.

$$-4\times \frac{-15y_{0}-4y_{0}^{2}}{-81-16y_{0}^{2}}+y_{0}$$

Express -4\times \frac{-15y_{0}-4y_{0}^{2}}{-81-16y_{0}^{2}} as a single fraction.

$$\frac{-4\left(-15y_{0}-4y_{0}^{2}\right)}{-81-16y_{0}^{2}}+y_{0}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply y_{0} times \frac{-81-16y_{0}^{2}}{-81-16y_{0}^{2}}.

$$\frac{-4\left(-15y_{0}-4y_{0}^{2}\right)}{-81-16y_{0}^{2}}+\frac{y_{0}\left(-81-16y_{0}^{2}\right)}{-81-16y_{0}^{2}}$$

Since \frac{-4\left(-15y_{0}-4y_{0}^{2}\right)}{-81-16y_{0}^{2}} and \frac{y_{0}\left(-81-16y_{0}^{2}\right)}{-81-16y_{0}^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-4\left(-15y_{0}-4y_{0}^{2}\right)+y_{0}\left(-81-16y_{0}^{2}\right)}{-81-16y_{0}^{2}}$$

Do the multiplications in -4\left(-15y_{0}-4y_{0}^{2}\right)+y_{0}\left(-81-16y_{0}^{2}\right).

$$\frac{60y_{0}+16y_{0}^{2}-81y_{0}-16y_{0}^{3}}{-81-16y_{0}^{2}}$$

Combine like terms in 60y_{0}+16y_{0}^{2}-81y_{0}-16y_{0}^{3}.

$$\frac{-21y_{0}+16y_{0}^{2}-16y_{0}^{3}}{-81-16y_{0}^{2}}$$

Expand

$$-\frac{-16y_{0}^{3}+16y_{0}^{2}-21y_{0}}{16y_{0}^{2}+81}$$

Short Solution Steps

Use the distributive property to multiply 27+4y_{0} by y_{0}.

$$-4\times \frac{12y_{0}-\left(27y_{0}+4y_{0}^{2}\right)}{27-4\left(27+4y_{0}^{2}\right)}+y_{0}$$

To find the opposite of 27y_{0}+4y_{0}^{2}, find the opposite of each term.

$$-4\times \frac{12y_{0}-27y_{0}-4y_{0}^{2}}{27-4\left(27+4y_{0}^{2}\right)}+y_{0}$$

Combine 12y_{0} and -27y_{0} to get -15y_{0}.

$$-4\times \frac{-15y_{0}-4y_{0}^{2}}{27-4\left(27+4y_{0}^{2}\right)}+y_{0}$$

Use the distributive property to multiply -4 by 27+4y_{0}^{2}.

$$-4\times \frac{-15y_{0}-4y_{0}^{2}}{27-108-16y_{0}^{2}}+y_{0}$$

Subtract 108 from 27 to get -81.

$$-4\times \frac{-15y_{0}-4y_{0}^{2}}{-81-16y_{0}^{2}}+y_{0}$$

Express -4\times \frac{-15y_{0}-4y_{0}^{2}}{-81-16y_{0}^{2}} as a single fraction.

$$\frac{-4\left(-15y_{0}-4y_{0}^{2}\right)}{-81-16y_{0}^{2}}+y_{0}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply y_{0} times \frac{-81-16y_{0}^{2}}{-81-16y_{0}^{2}}.

$$\frac{-4\left(-15y_{0}-4y_{0}^{2}\right)}{-81-16y_{0}^{2}}+\frac{y_{0}\left(-81-16y_{0}^{2}\right)}{-81-16y_{0}^{2}}$$

Since \frac{-4\left(-15y_{0}-4y_{0}^{2}\right)}{-81-16y_{0}^{2}} and \frac{y_{0}\left(-81-16y_{0}^{2}\right)}{-81-16y_{0}^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-4\left(-15y_{0}-4y_{0}^{2}\right)+y_{0}\left(-81-16y_{0}^{2}\right)}{-81-16y_{0}^{2}}$$

Do the multiplications in -4\left(-15y_{0}-4y_{0}^{2}\right)+y_{0}\left(-81-16y_{0}^{2}\right).

$$\frac{60y_{0}+16y_{0}^{2}-81y_{0}-16y_{0}^{3}}{-81-16y_{0}^{2}}$$

Combine like terms in 60y_{0}+16y_{0}^{2}-81y_{0}-16y_{0}^{3}.

$$\frac{-21y_{0}+16y_{0}^{2}-16y_{0}^{3}}{-81-16y_{0}^{2}}$$