#x^2+13\cdot x+42=# \((x+6)\cdot(x+7)\)

We are looking for numbers #p# and #q# such that the quadratic polynomial #x^2+13\cdot x+42# can be written as #(x-p)\cdot(x-q)#. If the absolute value of #p# is greater than #q#, we interchange them, so #|p|\le |q|#. We expand the brackets and compare the result with the original expression:
\[ x^2-(p+q)\cdot x+p\cdot q = x^2+13\cdot x+42\tiny\]
A comparison with #x^2+13\cdot x+42# gives \[
\lineqs{p+q &=& -13\cr p\cdot q &=& 42}\] If #p# and #q# are integers, they are divisors of #42#. We go through all possible divisors #p# with #p^2\le |42|# (which must be satisfied in view of #|p|\le |q|#) and in each case we calculate the sum of #p# and #q=\frac{42}{p}#:
\[\begin{array}{|r|c|l|}
\hline
p&q&{p+q}\\
\hline
1&42&43\\ \hline -1&-42&-43\\ \hline 2&21&23\\ \hline -2&-21&-23\\ \hline 3&14&17\\ \hline -3&-14&-17\\ \hline 6&7&13\\ \hline -6&-7&-13 \\
\hline
\end{array}\]
The line of the table with #p=-6# and #q=-7# is the only one with sum #-13#, hence, this is the answer:
\[x^2+13\cdot x+42=(x+6)\cdot(x+7)\tiny.\]