$$\frac{6x+7y+8}{2x+y}-\frac{2\left(x+13y+8\right)}{2\left(2x+y\right)}+6$$
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 $5x-6y+12x+6y$.$$\frac{17x}{2x+y}$$