Multiply both sides by the Least Common Denominator: \((x-2)(x+3)\).
\[(2x-5)(x+3)+{x}^{2}+1=(3x+8)(x-2)\]
Simplify.
\[3{x}^{2}+x-14=3{x}^{2}+2x-16\]
Cancel \(3{x}^{2}\) on both sides.
\[x-14=2x-16\]
Subtract \(x\) from both sides.
\[-14=2x-16-x\]
Simplify \(2x-16-x\) to \(x-16\).
\[-14=x-16\]
Add \(16\) to both sides.
\[-14+16=x\]
Simplify \(-14+16\) to \(2\).
\[2=x\]
Switch sides.
\[x=2\]
Check solution
When \(x=2\), the original equation \(\frac{2x-5}{x-2}+\frac{{x}^{2}+1}{{x}^{2}+x-6}=\frac{3x+8}{x+3}\) does not hold true.We will drop \(x=2\) from the solution set.
