Factor polynomial $$ -25x^{2}+65x-42 $$

$$ -25x^{2}+65x-42 = -( 5x-7 )( 5x-6 ) $$

Step 1 :

After factoring out $ -1 $ we have:

$$ -25x^{2}+65x-42 = - ~ ( 25x^{2}-65x+42 ) $$

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:

$$ a = 25 , b = -65 ~ \text{ and } ~ c = 42 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 25 }$ by the constant term $\color{blue}{c = 42} $.

$$ a \cdot c = 1050 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 1050 $ and add to $ b = -65 $.

Step 5: All pairs of numbers with a product of $ 1050 $ are:

PRODUCT = 1050
1     1050	-1     -1050
2     525	-2     -525
3     350	-3     -350
5     210	-5     -210
6     175	-6     -175
7     150	-7     -150
10     105	-10     -105
14     75	-14     -75
15     70	-15     -70
21     50	-21     -50
25     42	-25     -42
30     35	-30     -35

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -65 }$

PRODUCT = 1050 and SUM = -65
1     1050	-1     -1050
2     525	-2     -525
3     350	-3     -350
5     210	-5     -210
6     175	-6     -175
7     150	-7     -150
10     105	-10     -105
14     75	-14     -75
15     70	-15     -70
21     50	-21     -50
25     42	-25     -42
30     35	-30     -35

Step 7: Replace middle term $ -65 x $ with $ -30x-35x $:

$$ 25x^{2}-65x+42 = 25x^{2}-30x-35x+42 $$

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

$$ 25x^{2}-30x-35x+42 = 5x\left(5x-6\right) -7\left(5x-6\right) = \left(5x-7\right) \left(5x-6\right) $$

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