Factor polynomial $$ 5x^{2}-32x+35 $$

$$ 5x^{2}-32x+35 = ( x-5 )( 5x-7 ) $$

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:

$$ a = 5 , b = -32 ~ \text{ and } ~ c = 35 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = 35} $.

$$ a \cdot c = 175 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 175 $ and add to $ b = -32 $.

Step 4: All pairs of numbers with a product of $ 175 $ are:

PRODUCT = 175
1     175	-1     -175
5     35	-5     -35
7     25	-7     -25

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -32 }$

PRODUCT = 175 and SUM = -32
1     175	-1     -175
5     35	-5     -35
7     25	-7     -25

Step 6: Replace middle term $ -32 x $ with $ -7x-25x $:

$$ 5x^{2}-32x+35 = 5x^{2}-7x-25x+35 $$

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

$$ 5x^{2}-7x-25x+35 = x\left(5x-7\right) -5\left(5x-7\right) = \left(x-5\right) \left(5x-7\right) $$

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