MHB Factor $x^2-24x-17280$ Quickly & Effectively

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The quadratic expression $x^2-24x-17280$ can be factored as $(x-144)(x+120)$. To find the factors efficiently, the prime factorization of 17280 is utilized, which is $2^7 \cdot 3^3 \cdot 5$. The goal is to identify two factors that sum to -24, leading to the close values of 120 and 144. This method minimizes trial and error by focusing on the prime factors. The discussion highlights a systematic approach to factoring quadratics effectively.
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help factor this out using a faster and effective way.

$x^2-24x-17280$

I know that the factored form is $(x-144)(x+120)$

when I solved this I'm having a hard time finding a product of two numbers that will give me the middle term. can you give some fast way to determine that? with much less use of repetitious trial and error. thanks!
 
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I would first look at the prime factorization:

$$17280=2^7\cdot3^3\cdot5$$

Now, we want to find two factors whose sum is $-24$, so we know the two factors will need to be close in value. So, we could try:

$$17280=\left(2^3\cdot3\cdot5\right)\left(2^3\cdot3\cdot6\right)=120\cdot144$$

And this turns out to be the factors we need. :D
 
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