1. **State the problem:** We want to expand the expression $ (x+1)^3 $. 2. **Formula used:** The cube of a binomial $(a+b)^3$ is expanded using the formula: $$ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $$ 3. **Apply the formula:** Here, $a = x$ and $b = 1$. Substitute these values: $$ (x+1)^3 = x^3 + 3x^2(1) + 3x(1)^2 + 1^3 $$ 4. **Simplify each term:** $$ x^3 + 3x^2 + 3x + 1 $$ 5. **Final answer:** $$ (x+1)^3 = x^3 + 3x^2 + 3x + 1 $$ This expansion shows how the cube of a binomial is expressed as a polynomial with terms involving powers of $x$ and constants.