1. **State the problem:** We need to find the value of $v$ when $v = 2c^2 + 6$ and $c = 4$. 2. **Write the formula:** The formula given is $$v = 2c^2 + 6$$ 3. **Substitute the value of $c$:** Replace $c$ with 4 in the formula: $$v = 2 \times 4^2 + 6$$ 4. **Calculate the square:** $$v = 2 \times 16 + 6$$ 5. **Multiply:** $$v = 32 + 6$$ 6. **Add:** $$v = 38$$ **Final answer:** $v = 38$