In order to fill the table, we just need to calculate the value of the function for each value of x, so we have:
[tex]\begin{gathered} f(x)=x^2+2x+1 \\ x=-50\colon \\ f(-50)=(-50)^2+2\cdot(-50)+1=2500-100+1=2401 \\ x=-3\colon \\ f(-3)=(-3)^2+2\cdot(-3)+1=9-6+1=4 \\ x=-2\colon \\ f(-2)=(-2)^2+2\cdot(-2)+1=4-4+1=1 \\ x=-1\colon \\ f(-1)=(-1)^2+2\cdot(-1)+1=1-2+1=0_{} \\ x=0\colon \\ f(0)=0^2+2\cdot0+1=1 \\ x=1\colon \\ f(1)=1^2+2\cdot1+1=4 \\ x=2\colon \\ f(2)=2^2+2\cdot2+1=9 \\ x=3\colon \\ f(3)=3^2+2\cdot3+1=16 \\ x=50\colon \\ f(50)=50^2+2\cdot50+1=2601 \end{gathered}[/tex]