1. **State the problem:** Find the value(s) of $x$ that satisfy the equation $$\frac{x(x - 3)}{(x + 1)^2} = \frac{3}{5}$$ 2. **Formula and rules:** To solve this rational equation, we will cross-multiply to eliminate the denominator, then solve the resulting quadratic equation. Remember, $x \neq -1$ because it would make the denominator zero. 3. **Cross-multiply:** $$5 \cdot x(x - 3) = 3 \cdot (x + 1)^2$$ 4. **Expand both sides:** Left side: $$5(x^2 - 3x) = 5x^2 - 15x$$ Right side: $$(x + 1)^2 = x^2 + 2x + 1$$ So, $$3(x^2 + 2x + 1) = 3x^2 + 6x + 3$$ 5. **Set the equation:** $$5x^2 - 15x = 3x^2 + 6x + 3$$ 6. **Bring all terms to one side:** $$5x^2 - 15x - 3x^2 - 6x - 3 = 0$$ Simplify: $$2x^2 - 21x - 3 = 0$$ 7. **Solve the quadratic equation:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=-21$, $c=-3$. Calculate the discriminant: $$\Delta = (-21)^2 - 4 \cdot 2 \cdot (-3) = 441 + 24 = 465$$ 8. **Find the roots:** $$x = \frac{21 \pm \sqrt{465}}{4}$$ 9. **Check for restrictions:** $x \neq -1$ which is not equal to either root. **Final answer:** $$x = \frac{21 + \sqrt{465}}{4} \quad \text{or} \quad x = \frac{21 - \sqrt{465}}{4}$$