1. **State the question:** Why do we multiply both sides of the differential equation by the integrating factor? 2. **Recall the original equation:** $y' + P(x) y = Q(x)$. 3. **Purpose of the integrating factor:** The integrating factor $\mu(x) = e^{\int P(x) dx}$ is chosen so that when we multiply the entire equation by $\mu(x)$, the left side becomes the derivative of a product: $$\mu(x) y' + \mu(x) P(x) y = \frac{d}{dx} \left( \mu(x) y \right).$$ 4. **Why this helps:** This transformation allows us to rewrite the equation as: $$\frac{d}{dx} \left( \mu(x) y \right) = \mu(x) Q(x),$$ which can be integrated easily with respect to $x$. 5. **Summary:** Multiplying both sides by the integrating factor converts the left side into a single derivative, simplifying the process of solving the differential equation.