Factor polynomial $$ -7x^{2}+17x-10 $$

$$ -7x^{2}+17x-10 = -( 7x-10 )( x-1 ) $$

Step 1 :

After factoring out $ -1 $ we have:

$$ -7x^{2}+17x-10 = - ~ ( 7x^{2}-17x+10 ) $$

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:

$$ a = 7 , b = -17 ~ \text{ and } ~ c = 10 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 7 }$ by the constant term $\color{blue}{c = 10} $.

$$ a \cdot c = 70 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 70 $ and add to $ b = -17 $.

Step 5: All pairs of numbers with a product of $ 70 $ are:

PRODUCT = 70
1     70	-1     -70
2     35	-2     -35
5     14	-5     -14
7     10	-7     -10

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -17 }$

PRODUCT = 70 and SUM = -17
1     70	-1     -70
2     35	-2     -35
5     14	-5     -14
7     10	-7     -10

Step 7: Replace middle term $ -17 x $ with $ -7x-10x $:

$$ 7x^{2}-17x+10 = 7x^{2}-7x-10x+10 $$

Step 8: Apply factoring by grouping. Factor $ 7x $ out of the first two terms and $ -10 $ out of the last two terms.

$$ 7x^{2}-7x-10x+10 = 7x\left(x-1\right) -10\left(x-1\right) = \left(7x-10\right) \left(x-1\right) $$

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