1. **State the problem:** Simplify the expression $$A = (x+1)(x+x+1)(x-1)(x^2 - x + 1)$$. 2. **Rewrite and simplify terms:** Notice that $$x+x+1 = 2x+1$$, so the expression becomes $$A = (x+1)(2x+1)(x-1)(x^2 - x + 1)$$. 3. **Group and multiply the first two factors:** $$ (x+1)(2x+1) = 2x^2 + x + 2x + 1 = 2x^2 + 3x + 1 $$ 4. **Group and multiply the next two factors:** $$ (x-1)(x^2 - x + 1) $$ Multiply term by term: $$ x(x^2 - x + 1) - 1(x^2 - x + 1) = x^3 - x^2 + x - x^2 + x - 1 = x^3 - 2x^2 + 2x - 1 $$ 5. **Now multiply the two results:** $$ (2x^2 + 3x + 1)(x^3 - 2x^2 + 2x - 1) $$ Multiply each term: $$ 2x^2 \cdot x^3 = 2x^5 $$ $$ 2x^2 \cdot (-2x^2) = -4x^4 $$ $$ 2x^2 \cdot 2x = 4x^3 $$ $$ 2x^2 \cdot (-1) = -2x^2 $$ $$ 3x \cdot x^3 = 3x^4 $$ $$ 3x \cdot (-2x^2) = -6x^3 $$ $$ 3x \cdot 2x = 6x^2 $$ $$ 3x \cdot (-1) = -3x $$ $$ 1 \cdot x^3 = x^3 $$ $$ 1 \cdot (-2x^2) = -2x^2 $$ $$ 1 \cdot 2x = 2x $$ $$ 1 \cdot (-1) = -1 $$ 6. **Combine like terms:** $$ 2x^5 + (-4x^4 + 3x^4) + (4x^3 - 6x^3 + x^3) + (-2x^2 + 6x^2 - 2x^2) + (-3x + 2x) - 1 $$ Simplify: $$ 2x^5 - x^4 - x^3 + 2x^2 - x - 1 $$ **Final answer:** $$ A = 2x^5 - x^4 - x^3 + 2x^2 - x - 1 $$