Step 1 :

After factoring out $ 3 $ we have:

$$ 3x^{2}+45x+168 = 3 ( x^{2}+15x+56 ) $$

Step 2 :

Step 2: Identify constants $ \color{blue}{ b }$ and $\color{red}{ c }$. ( $ \color{blue}{ b }$ is a number in front of the $ x $ term and $ \color{red}{ c } $ is a constant). In our case:

$$ \color{blue}{ b = 15 } ~ \text{ and } ~ \color{red}{ c = 56 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 15 } $ and multiply to $ \color{red}{ 56 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 56 }$.

PRODUCT = 56
1     56	-1     -56
2     28	-2     -28
4     14	-4     -14
7     8	-7     -8

Step 4: Find out which pair sums up to $\color{blue}{ b = 15 }$

PRODUCT = 56 and SUM = 15
1     56	-1     -56
2     28	-2     -28
4     14	-4     -14
7     8	-7     -8

Step 5: Put 7 and 8 into placeholders to get factored form.

$$ \begin{aligned} x^{2}+15x+56 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+15x+56 & = (x + 7)(x + 8) \end{aligned} $$

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