Factor polynomial $$ 3x^{2}-57x+180 $$

$$ 3x^{2}-57x+180 = 3( x-4 )( x-15 ) $$

Step 1 :

After factoring out $ 3 $ we have:

$$ 3x^{2}-57x+180 = 3 ( x^{2}-19x+60 ) $$

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:

$$ \color{blue}{ b = -19 } ~ \text{ and } ~ \color{red}{ c = 60 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -19 } $ and multiply to $ \color{red}{ 60 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 60 }$.

PRODUCT = 60
1     60	-1     -60
2     30	-2     -30
3     20	-3     -20
4     15	-4     -15
5     12	-5     -12
6     10	-6     -10

Step 4: Find out which pair sums up to $\color{blue}{ b = -19 }$

PRODUCT = 60 and SUM = -19
1     60	-1     -60
2     30	-2     -30
3     20	-3     -20
4     15	-4     -15
5     12	-5     -12
6     10	-6     -10

Step 5: Put -4 and -15 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-19x+60 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-19x+60 & = (x -4)(x -15) \end{aligned} $$

Polynomial Factoring Calculator