Factor polynomial $$ 30a^{2}-57a+18 $$

$$ 30a^{2}-57a+18 = 3( 2a-3 )( 5a-2 ) $$

Step 1 :

After factoring out $ 3 $ we have:

$$ 30a^{2}-57a+18 = 3 ( 10a^{2}-19a+6 ) $$

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:

$$ a = 10 , b = -19 ~ \text{ and } ~ c = 6 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 10 }$ by the constant term $\color{blue}{c = 6} $.

$$ a \cdot c = 60 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 60 $ and add to $ b = -19 $.

Step 5: All pairs of numbers with a product of $ 60 $ are:

PRODUCT = 60
1     60	-1     -60
2     30	-2     -30
3     20	-3     -20
4     15	-4     -15
5     12	-5     -12
6     10	-6     -10

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -19 }$

PRODUCT = 60 and SUM = -19
1     60	-1     -60
2     30	-2     -30
3     20	-3     -20
4     15	-4     -15
5     12	-5     -12
6     10	-6     -10

Step 7: Replace middle term $ -19 x $ with $ -4x-15x $:

$$ 10x^{2}-19x+6 = 10x^{2}-4x-15x+6 $$

Step 8: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -3 $ out of the last two terms.

$$ 10x^{2}-4x-15x+6 = 2x\left(5x-2\right) -3\left(5x-2\right) = \left(2x-3\right) \left(5x-2\right) $$

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