Factor polynomial $$ 3n^{2}+10n-25 $$

$$ 3n^{2}+10n-25 = ( 3n-5 )( n+5 ) $$

Step 1: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:

$$ a = 3 , b = 10 ~ \text{ and } ~ c = -25 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = -25} $.

$$ a \cdot c = -75 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -75 $ and add to $ b = 10 $.

Step 4: All pairs of numbers with a product of $ -75 $ are:

PRODUCT = -75
-1     75	1     -75
-3     25	3     -25
-5     15	5     -15

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 10 }$

PRODUCT = -75 and SUM = 10
-1     75	1     -75
-3     25	3     -25
-5     15	5     -15

Step 6: Replace middle term $ 10 x $ with $ 15x-5x $:

$$ 3x^{2}+10x-25 = 3x^{2}+15x-5x-25 $$

Step 7: Apply factoring by grouping. Factor $ 3x $ out of the first two terms and $ -5 $ out of the last two terms.

$$ 3x^{2}+15x-5x-25 = 3x\left(x+5\right) -5\left(x+5\right) = \left(3x-5\right) \left(x+5\right) $$

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