To factor a polynomial of the form:
[tex]k^2+Bk+C[/tex]we need to find two integer numbers, a and b, that fullfil the following:
[tex]\begin{gathered} ab=C \\ a+b=B \end{gathered}[/tex]then we write the original polynomial as:
[tex]k^2+ak+bk+C[/tex]finally we factor by agrupation.
Let's do this with the polynomial:
[tex]k^2+8k+15[/tex]In this case we have that B=8 and C=15. We need two numbers which sum gives 8 and multiplication gives 15; this numbers can be a=5 and b=3, then:
[tex]\begin{gathered} 3\cdot5=15 \\ 3+5=8 \end{gathered}[/tex]Then we write the polynomial as:
[tex]k^2+5k+3k+15[/tex]and we factor by agrupation:
[tex]\begin{gathered} k^2+8k^2+15=k^2+5k+3k+15 \\ =k(k+5)+3(k+5) \\ =(k+3)(k+5) \end{gathered}[/tex]Therefore the factorization of the polynomial is:
[tex](k+3)(k+5)[/tex]