1. **State the problem:** Convert the quadratic expression $2x^2 + 4x + 1$ into its completing the square form. 2. **Recall the formula:** To complete the square for a quadratic expression $ax^2 + bx + c$, we first factor out $a$ from the $x^2$ and $x$ terms, then add and subtract the square of half the coefficient of $x$ inside the parentheses. 3. **Step-by-step solution:** - Start with the expression: $$2x^2 + 4x + 1$$ - Factor out 2 from the first two terms: $$2(x^2 + 2x) + 1$$ - Take half of the coefficient of $x$ inside the parentheses: half of 2 is 1. - Square it: $1^2 = 1$. - Add and subtract 1 inside the parentheses to complete the square: $$2(x^2 + 2x + 1 - 1) + 1$$ - Rewrite as: $$2((x + 1)^2 - 1) + 1$$ - Distribute 2: $$2(x + 1)^2 - 2 + 1$$ - Simplify constants: $$2(x + 1)^2 - 1$$ 4. **Final answer:** The expression in completing the square form is $$2(x + 1)^2 - 1$$. This form is useful for analyzing the vertex of the parabola represented by the quadratic expression.