Step 1: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:

$$ a = 3 , b = -17 ~ \text{ and } ~ c = -90 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = -90} $.

$$ a \cdot c = -270 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -270 $ and add to $ b = -17 $.

Step 4: All pairs of numbers with a product of $ -270 $ are:

PRODUCT = -270
-1     270	1     -270
-2     135	2     -135
-3     90	3     -90
-5     54	5     -54
-6     45	6     -45
-9     30	9     -30
-10     27	10     -27
-15     18	15     -18

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -17 }$

PRODUCT = -270 and SUM = -17
-1     270	1     -270
-2     135	2     -135
-3     90	3     -90
-5     54	5     -54
-6     45	6     -45
-9     30	9     -30
-10     27	10     -27
-15     18	15     -18

Step 6: Replace middle term $ -17 x $ with $ 10x-27x $:

$$ 3x^{2}-17x-90 = 3x^{2}+10x-27x-90 $$

Step 7: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -9 $ out of the last two terms.

$$ 3x^{2}+10x-27x-90 = x\left(3x+10\right) -9\left(3x+10\right) = \left(x-9\right) \left(3x+10\right) $$

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