Factor polynomial $$ 6x^{2}-35x+51 $$

$$ 6x^{2}-35x+51 = ( x-3 )( 6x-17 ) $$

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:

$$ a = 6 , b = -35 ~ \text{ and } ~ c = 51 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 6 }$ by the constant term $\color{blue}{c = 51} $.

$$ a \cdot c = 306 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 306 $ and add to $ b = -35 $.

Step 4: All pairs of numbers with a product of $ 306 $ are:

PRODUCT = 306
1     306	-1     -306
2     153	-2     -153
3     102	-3     -102
6     51	-6     -51
9     34	-9     -34
17     18	-17     -18

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -35 }$

PRODUCT = 306 and SUM = -35
1     306	-1     -306
2     153	-2     -153
3     102	-3     -102
6     51	-6     -51
9     34	-9     -34
17     18	-17     -18

Step 6: Replace middle term $ -35 x $ with $ -17x-18x $:

$$ 6x^{2}-35x+51 = 6x^{2}-17x-18x+51 $$

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

$$ 6x^{2}-17x-18x+51 = x\left(6x-17\right) -3\left(6x-17\right) = \left(x-3\right) \left(6x-17\right) $$

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