1. State the problem. Problem: Solve for $x$ in the equation $\frac{x^2 - 4}{x - 2} = 5$. 2. Formula and important rules. For rational equations of the form $\frac{A}{B}=C$ we multiply both sides by $B$ to remove denominators and then solve the resulting equation. Always state domain restrictions to avoid dividing by zero, so here $x \neq 2$ because the denominator $x-2$ cannot be zero. 3. Show intermediate work and factorization. Factor the numerator using difference of squares: $x^2 - 4 = (x-2)(x+2)$. Rewrite the equation as $$\frac{(x-2)(x+2)}{x-2} = 5$$ Show cancellation of the common factor explicitly: $$\frac{\cancel{(x-2)}(x+2)}{\cancel{(x-2)}} = 5$$ After cancelling the common factor we get the simpler equation $x+2 = 5$. Solve for $x$ to obtain $x = 3$. 4. Check and final answer. Substitute $x=3$ into the original equation to check: the left-hand side is $\frac{3^2-4}{3-2} = \frac{9-4}{1} = 5$ which equals the right-hand side, so $x=3$ is valid and not excluded by the domain restriction. Final answer: $x = 3$.