1. **State the problem:** Find the integral $$\int (5 - \frac{3}{x})^2 \, dx$$. 2. **Expand the integrand:** Use the formula $$(a - b)^2 = a^2 - 2ab + b^2$$ with $a=5$ and $b=\frac{3}{x}$: $$ (5 - \frac{3}{x})^2 = 5^2 - 2 \times 5 \times \frac{3}{x} + \left(\frac{3}{x}\right)^2 = 25 - \frac{30}{x} + \frac{9}{x^2} $$ 3. **Rewrite the integral:** $$ \int (5 - \frac{3}{x})^2 \, dx = \int \left(25 - \frac{30}{x} + \frac{9}{x^2}\right) dx $$ 4. **Integrate term-by-term:** - Integral of 25 is $$25x$$. - Integral of $$-\frac{30}{x}$$ is $$-30 \ln|x|$$. - Integral of $$\frac{9}{x^2} = 9x^{-2}$$ is: $$ \int 9x^{-2} dx = 9 \int x^{-2} dx = 9 \left(\frac{\cancel{x^{-1}}}{-1}\right) = -9x^{-1} = -\frac{9}{x} $$ 5. **Combine all results:** $$ \int (5 - \frac{3}{x})^2 dx = 25x - 30 \ln|x| - \frac{9}{x} + C $$ where $C$ is the constant of integration. **Final answer:** $$ \boxed{25x - 30 \ln|x| - \frac{9}{x} + C} $$