This method of proof is essentially directly showing the truth of a statement.
That is a chain of logic something of the form:
"if a is true then b is true".
"If b is true, then c is true."
"since a is true, then c must be true".
The function $k(n)$ can be rewritten as $$k(n) = n^2+ 4n + 7 = (n+2)^2 + 3$$
For all $n$, $(n+2)^2 \geq 0$ as squaring an integer gives either a positive answer or zero.
Therefore $(n+2)^2 + 3 \gt 0$ and hence $k(n)\gt 0$ for all $n$