Factor polynomial $$ 3x^{2}+51x+90 $$

$$ 3x^{2}+51x+90 = 3( x+2 )( x+15 ) $$

Step 1 :

After factoring out $ 3 $ we have:

$$ 3x^{2}+51x+90 = 3 ( x^{2}+17x+30 ) $$

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:

$$ \color{blue}{ b = 17 } ~ \text{ and } ~ \color{red}{ c = 30 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 17 } $ and multiply to $ \color{red}{ 30 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 30 }$.

PRODUCT = 30
1     30	-1     -30
2     15	-2     -15
3     10	-3     -10
5     6	-5     -6

Step 4: Find out which pair sums up to $\color{blue}{ b = 17 }$

PRODUCT = 30 and SUM = 17
1     30	-1     -30
2     15	-2     -15
3     10	-3     -10
5     6	-5     -6

Step 5: Put 2 and 15 into placeholders to get factored form.

$$ \begin{aligned} x^{2}+17x+30 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+17x+30 & = (x + 2)(x + 15) \end{aligned} $$

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