Evaluate: /(6 x+7 y+8) 2 x+y-/(2 x+26 y+16) 4 x+2 y+6

Expression: $$\frac { 6 x + 7 y + 8 } { 2 x + y } - \frac { 2 x + 26 y + 16 } { 4 x + 2 y } + 6$$

Factor the expressions that are not already factored in $\frac{2x+26y+16}{4x+2y}$.

$$\frac{6x+7y+8}{2x+y}-\frac{2\left(x+13y+8\right)}{2\left(2x+y\right)}+6$$

Cancel out $2$ in both numerator and denominator.

$$\frac{6x+7y+8}{2x+y}-\frac{x+13y+8}{2x+y}+6$$

Since $\frac{6x+7y+8}{2x+y}$ and $\frac{x+13y+8}{2x+y}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{6x+7y+8-\left(x+13y+8\right)}{2x+y}+6$$

Do the multiplications in $6x+7y+8-\left(x+13y+8\right)$.

$$\frac{6x+7y+8-x-13y-8}{2x+y}+6$$

Combine like terms in $6x+7y+8-x-13y-8$.

$$\frac{5x-6y}{2x+y}+6$$

To add or subtract expressions, expand them to make their denominators the same. Multiply $6$ times $\frac{2x+y}{2x+y}$.

$$\frac{5x-6y}{2x+y}+\frac{6\left(2x+y\right)}{2x+y}$$

Since $\frac{5x-6y}{2x+y}$ and $\frac{6\left(2x+y\right)}{2x+y}$ have the same denominator, add them by adding their numerators.

$$\frac{5x-6y+6\left(2x+y\right)}{2x+y}$$

Do the multiplications in $5x-6y+6\left(2x+y\right)$.

$$\frac{5x-6y+12x+6y}{2x+y}$$

Combine like terms in $5x-6y+12x+6y$.

$$\frac{17x}{2x+y}$$