Factoring Quadratics

A student wrongly factored a trinomial. To find the error, let's multiply the factors on the right-hand side to find the correct quadratic expression. We can see that the constant term above is $60,$ while the constant term in the given expression is $\text{-}60.$

Let's factor the expression by ourselves. Since the leading coefficient is $1,$ we need to find two numbers whose sum is $\text{-}17$ and whose product is $\text{-}60.$ Even though $\text{-}5$ and$\text{-}12$ add to $\text{-}17,$ these are not the correct numbers as their product is not $\text{-}60.$ Let's look at all the factors of $\text{-}60$ to see which ones have a sum of $\text{-}17.$

Factors	$1(\text{-}60)$	$2(\text{-}30)$	$3(\text{-}20)$	$4(\text{-}15)$	$5(\text{-}12)$	$6(\text{-}10)$
Sum	$\text{-}59$	$\text{-}28$	$\text{-}17$	$\text{-}11$	$\text{-}7$	$\text{-}4$

Since the sum of $3$ and $\text{-}20$ is $\text{-}17,$ and their product is $\text{-}60,$ these are the numbers we are looking for. With this information, we can correctly factor the given expression.