Step 1 :

After factoring out $ 3x^{2} $ we have:

$$ 3x^{4}+45x^{3}+42x^{2} = 3x^{2} ( x^{2}+15x+14 ) $$

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:

$$ \color{blue}{ b = 15 } ~ \text{ and } ~ \color{red}{ c = 14 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 15 } $ and multiply to $ \color{red}{ 14 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 14 }$.

PRODUCT = 14
1     14	-1     -14
2     7	-2     -7

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

PRODUCT = 14 and SUM = 15
1     14	-1     -14
2     7	-2     -7

Step 5: Put 1 and 14 into placeholders to get factored form.

$$ \begin{aligned} x^{2}+15x+14 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+15x+14 & = (x + 1)(x + 14) \end{aligned} $$

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