Factor polynomial $$ 4x^{2}-70x+250 $$

$$ 4x^{2}-70x+250 = 2( 2x-25 )( x-5 ) $$

Step 1 :

After factoring out $ 2 $ we have:

$$ 4x^{2}-70x+250 = 2 ( 2x^{2}-35x+125 ) $$

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:

$$ a = 2 , b = -35 ~ \text{ and } ~ c = 125 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = 125} $.

$$ a \cdot c = 250 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 250 $ and add to $ b = -35 $.

Step 5: All pairs of numbers with a product of $ 250 $ are:

PRODUCT = 250
1     250	-1     -250
2     125	-2     -125
5     50	-5     -50
10     25	-10     -25

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -35 }$

PRODUCT = 250 and SUM = -35
1     250	-1     -250
2     125	-2     -125
5     50	-5     -50
10     25	-10     -25

Step 7: Replace middle term $ -35 x $ with $ -10x-25x $:

$$ 2x^{2}-35x+125 = 2x^{2}-10x-25x+125 $$

Step 8: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -25 $ out of the last two terms.

$$ 2x^{2}-10x-25x+125 = 2x\left(x-5\right) -25\left(x-5\right) = \left(2x-25\right) \left(x-5\right) $$

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