Factor polynomial $$ -4x^{2}-61x-117 $$

$$ -4x^{2}-61x-117 = -( 4x+9 )( x+13 ) $$

Step 1 :

After factoring out $ -1 $ we have:

$$ -4x^{2}-61x-117 = - ~ ( 4x^{2}+61x+117 ) $$

Step 2 :

Step 2: 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:

$$ a = 4 , b = 61 ~ \text{ and } ~ c = 117 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = 117} $.

$$ a \cdot c = 468 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 468 $ and add to $ b = 61 $.

Step 5: All pairs of numbers with a product of $ 468 $ are:

PRODUCT = 468
1     468	-1     -468
2     234	-2     -234
3     156	-3     -156
4     117	-4     -117
6     78	-6     -78
9     52	-9     -52
12     39	-12     -39
13     36	-13     -36
18     26	-18     -26

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 61 }$

PRODUCT = 468 and SUM = 61
1     468	-1     -468
2     234	-2     -234
3     156	-3     -156
4     117	-4     -117
6     78	-6     -78
9     52	-9     -52
12     39	-12     -39
13     36	-13     -36
18     26	-18     -26

Step 7: Replace middle term $ 61 x $ with $ 52x+9x $:

$$ 4x^{2}+61x+117 = 4x^{2}+52x+9x+117 $$

Step 8: Apply factoring by grouping. Factor $ 4x $ out of the first two terms and $ 9 $ out of the last two terms.

$$ 4x^{2}+52x+9x+117 = 4x\left(x+13\right) + 9\left(x+13\right) = \left(4x+9\right) \left(x+13\right) $$

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