Factor polynomial $$ 50v^{2}-185v-180 $$

$$ 50v^{2}-185v-180 = 5( 2v-9 )( 5v+4 ) $$

Step 1 :

After factoring out $ 5 $ we have:

$$ 50v^{2}-185v-180 = 5 ( 10v^{2}-37v-36 ) $$

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:

$$ a = 10 , b = -37 ~ \text{ and } ~ c = -36 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 10 }$ by the constant term $\color{blue}{c = -36} $.

$$ a \cdot c = -360 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -360 $ and add to $ b = -37 $.

Step 5: All pairs of numbers with a product of $ -360 $ are:

PRODUCT = -360
-1     360	1     -360
-2     180	2     -180
-3     120	3     -120
-4     90	4     -90
-5     72	5     -72
-6     60	6     -60
-8     45	8     -45
-9     40	9     -40
-10     36	10     -36
-12     30	12     -30
-15     24	15     -24
-18     20	18     -20

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -37 }$

PRODUCT = -360 and SUM = -37
-1     360	1     -360
-2     180	2     -180
-3     120	3     -120
-4     90	4     -90
-5     72	5     -72
-6     60	6     -60
-8     45	8     -45
-9     40	9     -40
-10     36	10     -36
-12     30	12     -30
-15     24	15     -24
-18     20	18     -20

Step 7: Replace middle term $ -37 x $ with $ 8x-45x $:

$$ 10x^{2}-37x-36 = 10x^{2}+8x-45x-36 $$

Step 8: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -9 $ out of the last two terms.

$$ 10x^{2}+8x-45x-36 = 2x\left(5x+4\right) -9\left(5x+4\right) = \left(2x-9\right) \left(5x+4\right) $$

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