Step 1 :

After factoring out $ x $ we have:

$$ 7x^{3}-6x^{2}+51x = x ( 7x^{2}-6x+51 ) $$

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:

$$ a = 7 , b = -6 ~ \text{ and } ~ c = 51 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 7 }$ by the constant term $\color{blue}{c = 51} $.

$$ a \cdot c = 357 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 357 $ and add to $ b = -6 $.

Step 5: All pairs of numbers with a product of $ 357 $ are:

PRODUCT = 357
1     357	-1     -357
3     119	-3     -119
7     51	-7     -51
17     21	-17     -21

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

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ -6 }$, we conclude the polynomial cannot be factored.

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