Factor polynomial $$ 2a^{2}-13a+21 $$

$$ 2a^{2}-13a+21 = ( 2a-7 )( a-3 ) $$

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:

$$ a = 2 , b = -13 ~ \text{ and } ~ c = 21 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = 21} $.

$$ a \cdot c = 42 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 42 $ and add to $ b = -13 $.

Step 4: All pairs of numbers with a product of $ 42 $ are:

PRODUCT = 42
1     42	-1     -42
2     21	-2     -21
3     14	-3     -14
6     7	-6     -7

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -13 }$

PRODUCT = 42 and SUM = -13
1     42	-1     -42
2     21	-2     -21
3     14	-3     -14
6     7	-6     -7

Step 6: Replace middle term $ -13 x $ with $ -6x-7x $:

$$ 2x^{2}-13x+21 = 2x^{2}-6x-7x+21 $$

Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -7 $ out of the last two terms.

$$ 2x^{2}-6x-7x+21 = 2x\left(x-3\right) -7\left(x-3\right) = \left(2x-7\right) \left(x-3\right) $$

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