#### Answer

domain: $(-\infty,\ \infty)$
range: $[4,\infty)$

#### Work Step by Step

We want to find the domain and range of:
$F(t)=t^{2}+2t+5$
We factor the equation by completing the square:
$t^{2}+2t+5$
$(t^{2}+2t+1)+5-1$
$(t+1)^{2}+4$
We can see that this is a shifted parabola ($t^2$ with the trough at x=-1, y=4 opening up). We know that F(t) will be larger than 4 for any $t$. Thus the range is: $[4,\infty)$. We also know that $x$ can be any real number, so the domain is: $(-\infty,\ \infty)$.