Question: Consider $\Phi$ to be the space of sequences that have finitely many non-zero terms.
The space is not closed in $\ell\_1$, therefore $\ell\_1/\Phi$ with the quotient topology is not Hausdorff, and so it cannot be metrizable. However, does there exist a metric on $\ell\_1/\Phi$ that gives rise to a non-trivial topology. And even stronger question is, is $\ell\_1/\Phi$ normable?

Answer: 
I would use $$\sum\_{i=1} ^n (i+2)! - i! = (n+2)!+(n+1)! - 2! - 1 !$$
If you observe that $i^2 + 3i + 1 = (i + 2)(i+1) - 1$, then your left-hand sum is
$$\sum\_{i=1}^n ((i+2)(i +1) - 1)i!$$
$$= \sum\_{i=1}^n ((i + 2)! - i!)$$
As Henry noted, this sum telescopes into
$$(n + 2)! + (n + 1)! - 2! - 1!$$
$$= (n+2)(n+1)! + (n+1)! - 3$$
$$= (n+3)(n+1)! - 3$$