Factor polynomial $$ 4x^{2}+3x-27 $$

$$ 4x^{2}+3x-27 = ( 4x-9 )( x+3 ) $$

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:

$$ a = 4 , b = 3 ~ \text{ and } ~ c = -27 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = -27} $.

$$ a \cdot c = -108 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -108 $ and add to $ b = 3 $.

Step 4: All pairs of numbers with a product of $ -108 $ are:

PRODUCT = -108
-1     108	1     -108
-2     54	2     -54
-3     36	3     -36
-4     27	4     -27
-6     18	6     -18
-9     12	9     -12

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 3 }$

PRODUCT = -108 and SUM = 3
-1     108	1     -108
-2     54	2     -54
-3     36	3     -36
-4     27	4     -27
-6     18	6     -18
-9     12	9     -12

Step 6: Replace middle term $ 3 x $ with $ 12x-9x $:

$$ 4x^{2}+3x-27 = 4x^{2}+12x-9x-27 $$

Step 7: Apply factoring by grouping. Factor $ 4x $ out of the first two terms and $ -9 $ out of the last two terms.

$$ 4x^{2}+12x-9x-27 = 4x\left(x+3\right) -9\left(x+3\right) = \left(4x-9\right) \left(x+3\right) $$

Polynomial Factoring Calculator