Factor polynomial $$ 3x^{3}+15x^{2}+18x $$

$$ 3x^{3}+15x^{2}+18x = 3x( x+2 )( x+3 ) $$

Step 1 :

After factoring out $ 3x $ we have:

$$ 3x^{3}+15x^{2}+18x = 3x ( x^{2}+5x+6 ) $$

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:

$$ \color{blue}{ b = 5 } ~ \text{ and } ~ \color{red}{ c = 6 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 5 } $ and multiply to $ \color{red}{ 6 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 6 }$.

PRODUCT = 6
1     6	-1     -6
2     3	-2     -3

Step 4: Find out which pair sums up to $\color{blue}{ b = 5 }$

PRODUCT = 6 and SUM = 5
1     6	-1     -6
2     3	-2     -3

Step 5: Put 2 and 3 into placeholders to get factored form.

$$ \begin{aligned} x^{2}+5x+6 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+5x+6 & = (x + 2)(x + 3) \end{aligned} $$

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