1. **State the problem:** Factor completely the expression $$7(3x + 2)^2 (x - 1)^2 + (3x + 2)(x - 1)^3$$ and simplify as much as possible. 2. **Identify common factors:** Both terms contain factors of $(3x + 2)$ and $(x - 1)$ raised to powers. The first term has $(3x + 2)^2 (x - 1)^2$ and the second term has $(3x + 2)(x - 1)^3$. 3. **Find the lowest powers of common factors:** The lowest power of $(3x + 2)$ is 1 and the lowest power of $(x - 1)$ is 2. 4. **Factor out the common terms:** $$ 7(3x + 2)^2 (x - 1)^2 + (3x + 2)(x - 1)^3 = (3x + 2)^1 (x - 1)^2 \left(7(3x + 2)^1 + (x - 1)^1\right) $$ 5. **Simplify inside the parentheses:** $$ 7(3x + 2) + (x - 1) = 7 \times 3x + 7 \times 2 + x - 1 = 21x + 14 + x - 1 = 22x + 13 $$ 6. **Write the fully factored expression:** $$ (3x + 2)(x - 1)^2 (22x + 13) $$ This is the completely factored and simplified form. **Final answer:** $$ (3x + 2)(x - 1)^2 (22x + 13) $$