Factor polynomial $$ -25x^{3}-340x^{2}+525x $$

$$ -25x^{3}-340x^{2}+525x = ( -5 )x( 5x-7 )( x+15 ) $$

Step 1 :

After factoring out $ -5x $ we have:

$$ -25x^{3}-340x^{2}+525x = -5x ( 5x^{2}+68x-105 ) $$

Step 2 :

Step 2: 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:

$$ a = 5 , b = 68 ~ \text{ and } ~ c = -105 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = -105} $.

$$ a \cdot c = -525 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -525 $ and add to $ b = 68 $.

Step 5: All pairs of numbers with a product of $ -525 $ are:

PRODUCT = -525
-1     525	1     -525
-3     175	3     -175
-5     105	5     -105
-7     75	7     -75
-15     35	15     -35
-21     25	21     -25

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 68 }$

PRODUCT = -525 and SUM = 68
-1     525	1     -525
-3     175	3     -175
-5     105	5     -105
-7     75	7     -75
-15     35	15     -35
-21     25	21     -25

Step 7: Replace middle term $ 68 x $ with $ 75x-7x $:

$$ 5x^{2}+68x-105 = 5x^{2}+75x-7x-105 $$

Step 8: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ -7 $ out of the last two terms.

$$ 5x^{2}+75x-7x-105 = 5x\left(x+15\right) -7\left(x+15\right) = \left(5x-7\right) \left(x+15\right) $$

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