Factor polynomial $$ 9x^{2}+25x-6 $$

$$ 9x^{2}+25x-6 = ( 9x-2 )( x+3 ) $$

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:

$$ a = 9 , b = 25 ~ \text{ and } ~ c = -6 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 9 }$ by the constant term $\color{blue}{c = -6} $.

$$ a \cdot c = -54 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -54 $ and add to $ b = 25 $.

Step 4: All pairs of numbers with a product of $ -54 $ are:

PRODUCT = -54
-1     54	1     -54
-2     27	2     -27
-3     18	3     -18
-6     9	6     -9

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 25 }$

PRODUCT = -54 and SUM = 25
-1     54	1     -54
-2     27	2     -27
-3     18	3     -18
-6     9	6     -9

Step 6: Replace middle term $ 25 x $ with $ 27x-2x $:

$$ 9x^{2}+25x-6 = 9x^{2}+27x-2x-6 $$

Step 7: Apply factoring by grouping. Factor $ 9x $ out of the first two terms and $ -2 $ out of the last two terms.

$$ 9x^{2}+27x-2x-6 = 9x\left(x+3\right) -2\left(x+3\right) = \left(9x-2\right) \left(x+3\right) $$

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