Step 1 :

After factoring out $ 5 $ we have:

$$ 5r^{2}+85r+300 = 5 ( r^{2}+17r+60 ) $$

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:

$$ \color{blue}{ b = 17 } ~ \text{ and } ~ \color{red}{ c = 60 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 17 } $ 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 $\color{blue}{ b = 17 }$

PRODUCT = 60 and SUM = 17
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 5 and 12 into placeholders to get factored form.

$$ \begin{aligned} r^{2}+17r+60 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ r^{2}+17r+60 & = (x + 5)(x + 12) \end{aligned} $$

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