$\begin{array} { l }\frac{ 4 }{ x-5 }+\frac{ 4 }{ x-6 }=\frac{ {x}^{2}-37 }{ {x}^{2}-11x+30 },& \begin{array} { l }x≠5,& x≠6\end{array}\end{array}$
Move the expression to the left-hand side and change its sign$\frac{ 4 }{ x-5 }+\frac{ 4 }{ x-6 }-\frac{ {x}^{2}-37 }{ {x}^{2}-11x+30 }=0$
Write $-11x$ as a difference$\frac{ 4 }{ x-5 }+\frac{ 4 }{ x-6 }-\frac{ {x}^{2}-37 }{ {x}^{2}-5x-6x+30 }=0$
Factor out $x$ from the expression$\frac{ 4 }{ x-5 }+\frac{ 4 }{ x-6 }-\frac{ {x}^{2}-37 }{ x \times \left( x-5 \right)-6x+30 }=0$
Factor out $-6$ from the expression$\frac{ 4 }{ x-5 }+\frac{ 4 }{ x-6 }-\frac{ {x}^{2}-37 }{ x \times \left( x-5 \right)-6\left( x-5 \right) }=0$
Factor out $x-5$ from the expression$\frac{ 4 }{ x-5 }+\frac{ 4 }{ x-6 }-\frac{ {x}^{2}-37 }{ \left( x-5 \right) \times \left( x-6 \right) }=0$
Write all numerators above the least common denominator $\left( x-5 \right) \times \left( x-6 \right)$$\frac{ 4\left( x-6 \right)+4\left( x-5 \right)-\left( {x}^{2}-37 \right) }{ \left( x-5 \right) \times \left( x-6 \right) }=0$
Distribute $4$ through the parentheses$\frac{ 4x-24+4\left( x-5 \right)-\left( {x}^{2}-37 \right) }{ \left( x-5 \right) \times \left( x-6 \right) }=0$
Distribute $4$ through the parentheses$\frac{ 4x-24+4x-20-\left( {x}^{2}-37 \right) }{ \left( x-5 \right) \times \left( x-6 \right) }=0$
When there is a $-$ in front of an expression in parentheses, change the sign of each term of the expression and remove the parentheses$\frac{ 4x-24+4x-20-{x}^{2}+37 }{ \left( x-5 \right) \times \left( x-6 \right) }=0$
Collect like terms$\frac{ 8x-24-20-{x}^{2}+37 }{ \left( x-5 \right) \times \left( x-6 \right) }=0$
Calculate the sum or difference$\frac{ 8x-7-{x}^{2} }{ \left( x-5 \right) \times \left( x-6 \right) }=0$
When the quotient of expressions equals $0$, the numerator has to be $0$$8x-7-{x}^{2}=0$
Use the commutative property to reorder the terms$-{x}^{2}+8x-7=0$
Change the signs on both sides of the equation${x}^{2}-8x+7=0$
Write $-8x$ as a difference${x}^{2}-x-7x+7=0$
Factor out $x$ from the expression$x \times \left( x-1 \right)-7x+7=0$
Factor out $-7$ from the expression$x \times \left( x-1 \right)-7\left( x-1 \right)=0$
Factor out $x-1$ from the expression$\left( x-1 \right) \times \left( x-7 \right)=0$
When the product of factors equals $0$, at least one factor is $0$$\begin{array} { l }x-1=0,\\x-7=0\end{array}$
Solve the equation for $x$$\begin{array} { l }x=1,\\x-7=0\end{array}$
Solve the equation for $x$$\begin{array} { l }\begin{array} { l }x=1,\\x=7\end{array},& \begin{array} { l }x≠5,& x≠6\end{array}\end{array}$
Check if the solution is in the defined range$\begin{array} { l }x=1,\\x=7\end{array}$
The equation has $2$ solutions$\begin{array} { l }x_1=1,& x_2=7\end{array}$