Factor polynomial $$ 8x^{2}-50x+63 $$

$$ 8x^{2}-50x+63 = ( 2x-9 )( 4x-7 ) $$

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:

$$ a = 8 , b = -50 ~ \text{ and } ~ c = 63 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 8 }$ by the constant term $\color{blue}{c = 63} $.

$$ a \cdot c = 504 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 504 $ and add to $ b = -50 $.

Step 4: All pairs of numbers with a product of $ 504 $ are:

PRODUCT = 504
1     504	-1     -504
2     252	-2     -252
3     168	-3     -168
4     126	-4     -126
6     84	-6     -84
7     72	-7     -72
8     63	-8     -63
9     56	-9     -56
12     42	-12     -42
14     36	-14     -36
18     28	-18     -28
21     24	-21     -24

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -50 }$

PRODUCT = 504 and SUM = -50
1     504	-1     -504
2     252	-2     -252
3     168	-3     -168
4     126	-4     -126
6     84	-6     -84
7     72	-7     -72
8     63	-8     -63
9     56	-9     -56
12     42	-12     -42
14     36	-14     -36
18     28	-18     -28
21     24	-21     -24

Step 6: Replace middle term $ -50 x $ with $ -14x-36x $:

$$ 8x^{2}-50x+63 = 8x^{2}-14x-36x+63 $$

Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -9 $ out of the last two terms.

$$ 8x^{2}-14x-36x+63 = 2x\left(4x-7\right) -9\left(4x-7\right) = \left(2x-9\right) \left(4x-7\right) $$

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