1. **State the problem:** We need to find the value of $a$ in the equation $$a = \frac{b^2 - 6}{c^2 - 2}$$ when $b = 4$ and $c = 2$. 2. **Write the formula:** The formula given is $$a = \frac{b^2 - 6}{c^2 - 2}$$. 3. **Substitute the values:** Substitute $b = 4$ and $c = 2$ into the formula: $$a = \frac{4^2 - 6}{2^2 - 2}$$ 4. **Calculate powers:** Calculate $4^2$ and $2^2$: $$a = \frac{16 - 6}{4 - 2}$$ 5. **Simplify numerator and denominator:** $$a = \frac{10}{2}$$ 6. **Simplify the fraction:** $$a = \frac{\cancel{10}}{\cancel{2}} = 5$$ 7. **Final answer:** $$\boxed{5}$$