#x^2+3\cdot x-40=# \((x-5)\cdot(x+8)\)

We are looking for numbers #p# and #q# such that the quadratic polynomial #x^2+3\cdot x-40# 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:
\[ x^2-(p+q)\cdot x+p\cdot q = x^2+3\cdot x-40\tiny\]
A comparison with #x^2+3\cdot x-40# gives \[
\lineqs{p+q &=& -3\cr p\cdot q &=& -40}\] If #p# and #q# are integers, they are divisors of #-40#. We go through all possible divisors #p# with #p^2\le |-40|# (which must be satisfied in view of #|p|\le |q|#) and in each case we calculate the sum of #p# and #q=\frac{-40}{p}#:
\[\begin{array}{|r|c|l|}
\hline
p&q&{p+q}\\
\hline
1&-40&-39\\ \hline -1&40&39\\ \hline 2&-20&-18\\ \hline -2&20&18\\ \hline 4&-10&-6\\ \hline -4&10&6\\ \hline 5&-8&-3\\ \hline -5&8&3 \\
\hline
\end{array}\]
The line of the table with #p=5# and #q=-8# is the only one with sum #-3#, hence, this is the answer:
\[x^2+3\cdot x-40=(x-5)\cdot(x+8)\tiny.\]