1. **State the problem:** Solve the exponential equation $$\left(\sqrt{3} - 1\right)^{5x+1} = \left( \frac{\sqrt{3} + 1}{2} \right)^4.$$\n\n2. **Recall important rules:** To solve equations with exponents, try to express both sides with the same base if possible. Then, equate the exponents.\n\n3. **Rewrite the bases:** Note that $$\sqrt{3} - 1$$ and $$\frac{\sqrt{3} + 1}{2}$$ are related. In fact, $$\frac{\sqrt{3} + 1}{2} = \frac{1}{\sqrt{3} - 1}$$ because multiplying numerator and denominator by $$\sqrt{3} - 1$$ gives 1 in the denominator.\n\n4. **Express right side base as reciprocal:** $$\left( \frac{\sqrt{3} + 1}{2} \right)^4 = \left( \frac{1}{\sqrt{3} - 1} \right)^4 = (\sqrt{3} - 1)^{-4}.$$\n\n5. **Rewrite the equation:** $$ (\sqrt{3} - 1)^{5x+1} = (\sqrt{3} - 1)^{-4}.$$\n\n6. **Equate exponents:** Since the bases are the same and nonzero, set exponents equal: $$5x + 1 = -4.$$\n\n7. **Solve for x:** $$5x = -4 - 1 = -5 \implies x = \frac{-5}{5} = -1.$$\n\n**Final answer:** $$x = -1.$$