Factor polynomial $$ 30k^{2}-171k+162 $$

$$ 30k^{2}-171k+162 = 3( 2k-9 )( 5k-6 ) $$

Step 1 :

After factoring out $ 3 $ we have:

$$ 30k^{2}-171k+162 = 3 ( 10k^{2}-57k+54 ) $$

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:

$$ a = 10 , b = -57 ~ \text{ and } ~ c = 54 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 10 }$ by the constant term $\color{blue}{c = 54} $.

$$ a \cdot c = 540 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 540 $ and add to $ b = -57 $.

Step 5: All pairs of numbers with a product of $ 540 $ are:

PRODUCT = 540
1     540	-1     -540
2     270	-2     -270
3     180	-3     -180
4     135	-4     -135
5     108	-5     -108
6     90	-6     -90
9     60	-9     -60
10     54	-10     -54
12     45	-12     -45
15     36	-15     -36
18     30	-18     -30
20     27	-20     -27

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -57 }$

PRODUCT = 540 and SUM = -57
1     540	-1     -540
2     270	-2     -270
3     180	-3     -180
4     135	-4     -135
5     108	-5     -108
6     90	-6     -90
9     60	-9     -60
10     54	-10     -54
12     45	-12     -45
15     36	-15     -36
18     30	-18     -30
20     27	-20     -27

Step 7: Replace middle term $ -57 x $ with $ -12x-45x $:

$$ 10x^{2}-57x+54 = 10x^{2}-12x-45x+54 $$

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

$$ 10x^{2}-12x-45x+54 = 2x\left(5x-6\right) -9\left(5x-6\right) = \left(2x-9\right) \left(5x-6\right) $$

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