1. **State the problem:** Solve for $a$ in the equation $$\frac{a - 2}{4a + 2} = \frac{1}{2}.$$\n\n2. **Use the cross-multiplication rule:** When two fractions are equal, their cross products are equal. So, $$2(a - 2) = 1(4a + 2).$$\n\n3. **Expand both sides:** $$2a - 4 = 4a + 2.$$\n\n4. **Bring all terms involving $a$ to one side and constants to the other:** $$2a - 4a = 2 + 4.$$\n\n5. **Simplify:** $$\cancel{2a} - 4a = 6 \implies -2a = 6.$$\n\n6. **Divide both sides by $-2$ to isolate $a$:** $$a = \frac{6}{\cancel{-2}}\cancel{-1} = -3.$$\n\n**Final answer:** $$a = -3.$$