1. **State the problem:** Given vectors $\mathbf{u} = (0, 4)$ and $\mathbf{v} = (3, 3)$, find the vector $4\mathbf{u} + 2\mathbf{v}$. 2. **Recall the formula:** Scalar multiplication and vector addition are performed component-wise. For vectors $\mathbf{a} = (a_1, a_2)$ and $\mathbf{b} = (b_1, b_2)$, and scalars $c, d$, $$c\mathbf{a} + d\mathbf{b} = (c a_1 + d b_1, c a_2 + d b_2)$$ 3. **Calculate each scalar multiplication:** $$4\mathbf{u} = 4(0,4) = (4 \times 0, 4 \times 4) = (0, 16)$$ $$2\mathbf{v} = 2(3,3) = (2 \times 3, 2 \times 3) = (6, 6)$$ 4. **Add the resulting vectors:** $$4\mathbf{u} + 2\mathbf{v} = (0, 16) + (6, 6) = (0 + 6, 16 + 6) = (6, 22)$$ 5. **Final answer:** The vector $4\mathbf{u} + 2\mathbf{v}$ is $(6, 22)$, which corresponds to option A.