1. **State the problem:** Solve for $x$ given the function $f(x) = \frac{x^2 - 5}{x - 2}$ and $f(x) = 4$. 2. **Write the equation:** $$\frac{x^2 - 5}{x - 2} = 4$$ 3. **Multiply both sides by the denominator to clear the fraction:** $$x^2 - 5 = 4(x - 2)$$ 4. **Expand the right side:** $$x^2 - 5 = 4x - 8$$ 5. **Bring all terms to one side to set the equation to zero:** $$x^2 - 5 - 4x + 8 = 0$$ 6. **Simplify:** $$x^2 - 4x + 3 = 0$$ 7. **Factor the quadratic:** $$ (x - 3)(x - 1) = 0 $$ 8. **Set each factor equal to zero and solve:** $$x - 3 = 0 \Rightarrow x = 3$$ $$x - 1 = 0 \Rightarrow x = 1$$ 9. **Check for restrictions:** The denominator $x - 2 \neq 0$, so $x \neq 2$. Both $x=1$ and $x=3$ are valid. **Final answer:** $$x = 1 \text{ or } x = 3$$