Solve y-4/4y^2+7y-2-y+2/8y^2+10y-3-4y+1/2y^2+7y+6

$$\frac { y - 4 } { 4 y ^ { 2 } + 7 y - 2 } - \frac { y + 2 } { 8 y ^ { 2 } + 10 y - 3 } - \frac { 4 y + 1 } { 2 y ^ { 2 } + 7 y + 6 }$$

Evaluate

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

Short Solution Steps

Factor 4y^{2}+7y-2. Factor 8y^{2}+10y-3.

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

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

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

Since \frac{\left(y-4\right)\left(2y+3\right)}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)} and \frac{\left(y+2\right)\left(y+2\right)}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(y-4\right)\left(2y+3\right)-\left(y+2\right)\left(y+2\right).

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

Combine like terms in 2y^{2}+3y-8y-12-y^{2}-2y-2y-4.

$$\frac{y^{2}-9y-16}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)}-\frac{4y+1}{2y^{2}+7y+6}$$

Factor 2y^{2}+7y+6.

$$\frac{y^{2}-9y-16}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)}-\frac{4y+1}{\left(y+2\right)\left(2y+3\right)}$$

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

$$\frac{y^{2}-9y-16}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)}-\frac{\left(4y+1\right)\left(4y-1\right)}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)}$$

Since \frac{y^{2}-9y-16}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)} and \frac{\left(4y+1\right)\left(4y-1\right)}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{y^{2}-9y-16-\left(4y+1\right)\left(4y-1\right)}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)}$$

Do the multiplications in y^{2}-9y-16-\left(4y+1\right)\left(4y-1\right).

$$\frac{y^{2}-9y-16-16y^{2}+4y-4y+1}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)}$$

Combine like terms in y^{2}-9y-16-16y^{2}+4y-4y+1.

$$\frac{-15y^{2}-9y-15}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)}$$

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

$$\frac{-15y^{2}-9y-15}{8y^{3}+26y^{2}+17y-6}$$

Expand

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

Short Solution Steps

Factor 4y^{2}+7y-2. Factor 8y^{2}+10y-3.

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

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

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

Since \frac{\left(y-4\right)\left(2y+3\right)}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)} and \frac{\left(y+2\right)\left(y+2\right)}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(y-4\right)\left(2y+3\right)-\left(y+2\right)\left(y+2\right).

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

Combine like terms in 2y^{2}+3y-8y-12-y^{2}-2y-2y-4.

$$\frac{y^{2}-9y-16}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)}-\frac{4y+1}{2y^{2}+7y+6}$$

Factor 2y^{2}+7y+6.

$$\frac{y^{2}-9y-16}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)}-\frac{4y+1}{\left(y+2\right)\left(2y+3\right)}$$

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

$$\frac{y^{2}-9y-16}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)}-\frac{\left(4y+1\right)\left(4y-1\right)}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)}$$

Since \frac{y^{2}-9y-16}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)} and \frac{\left(4y+1\right)\left(4y-1\right)}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{y^{2}-9y-16-\left(4y+1\right)\left(4y-1\right)}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)}$$

Do the multiplications in y^{2}-9y-16-\left(4y+1\right)\left(4y-1\right).

$$\frac{y^{2}-9y-16-16y^{2}+4y-4y+1}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)}$$

Combine like terms in y^{2}-9y-16-16y^{2}+4y-4y+1.

$$\frac{-15y^{2}-9y-15}{\left(4y-1\right)\left(y+2\right)\left(2y+3\right)}$$

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

$$\frac{-15y^{2}-9y-15}{8y^{3}+26y^{2}+17y-6}$$