\[ F(x)=x^{2}+2 x+1 \] Find the range of the quadratic Functios

Step 1 ：Rewrite the function $F(x)=x^{2}+2 x+1$ in vertex form as $F(x)=(x+1)^2$

Step 2 ：Identify the vertex of the parabola as $(-1,0)$

Step 3 ：Since the coefficient of $x^2$ is positive, the parabola opens upwards, meaning the vertex is the minimum point of the parabola

Step 4 ：Therefore, the range of the function is $y\geq0$ or in interval notation, $[0, \infty)$

Step 5 ：\(\boxed{[0, \infty)}\) is the final answer