Factor polynomial $$ 9v^{2}-42v+40 $$

$$ 9v^{2}-42v+40 = ( 3v-10 )( 3v-4 ) $$

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:

$$ a = 9 , b = -42 ~ \text{ and } ~ c = 40 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 9 }$ by the constant term $\color{blue}{c = 40} $.

$$ a \cdot c = 360 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 360 $ and add to $ b = -42 $.

Step 4: 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 5: Find out which factor pair sums up to $\color{blue}{ b = -42 }$

PRODUCT = 360 and SUM = -42
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: Replace middle term $ -42 x $ with $ -12x-30x $:

$$ 9x^{2}-42x+40 = 9x^{2}-12x-30x+40 $$

Step 7: Apply factoring by grouping. Factor $ 3x $ out of the first two terms and $ -10 $ out of the last two terms.

$$ 9x^{2}-12x-30x+40 = 3x\left(3x-4\right) -10\left(3x-4\right) = \left(3x-10\right) \left(3x-4\right) $$

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