To factor a quadratic polynomial of the form:
[tex]n^2+bn+c[/tex]we need to find to intergers B and C that fulfill the following conditions:
[tex]\begin{gathered} BC=c \\ \text{and} \\ B+C=b \end{gathered}[/tex]In the case of the polynomial:
[tex]n^2-4n-32[/tex]we notice that b=-4 and c=-32. Then we need to find two numbers that fulfills:
[tex]\begin{gathered} -32=BC \\ -4=B+C \end{gathered}[/tex]if we choose B=-8 and C=4 we notice that this requierements are fulfill. Once we have this numbers we write the polynomial as:
[tex]n^2-8n+4n-32[/tex]and we factor the first two terms and the last two terms by common factors:
[tex]\begin{gathered} n^2-4n-32=n^2-8n+4n-32=n(n-8)+4(n-8) \\ =(n-8)(n+4) \end{gathered}[/tex]Therefore:
[tex]n^2-4n-32=(n-8)(n+4)[/tex]