Factor polynomial $$ -4x^{2}+31x-21 $$

$$ -4x^{2}+31x-21 = -( x-7 )( 4x-3 ) $$

Step 1 :

After factoring out $ -1 $ we have:

$$ -4x^{2}+31x-21 = - ~ ( 4x^{2}-31x+21 ) $$

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:

$$ a = 4 , b = -31 ~ \text{ and } ~ c = 21 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = 21} $.

$$ a \cdot c = 84 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 84 $ and add to $ b = -31 $.

Step 5: All pairs of numbers with a product of $ 84 $ are:

PRODUCT = 84
1     84	-1     -84
2     42	-2     -42
3     28	-3     -28
4     21	-4     -21
6     14	-6     -14
7     12	-7     -12

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -31 }$

PRODUCT = 84 and SUM = -31
1     84	-1     -84
2     42	-2     -42
3     28	-3     -28
4     21	-4     -21
6     14	-6     -14
7     12	-7     -12

Step 7: Replace middle term $ -31 x $ with $ -3x-28x $:

$$ 4x^{2}-31x+21 = 4x^{2}-3x-28x+21 $$

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

$$ 4x^{2}-3x-28x+21 = x\left(4x-3\right) -7\left(4x-3\right) = \left(x-7\right) \left(4x-3\right) $$

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