Factor polynomial $$ -56x^{7}+32x^{6}-48x^{5} $$

$$ -56x^{7}+32x^{6}-48x^{5} = ( -8 )x^{5}( 7x^{2}-4x+6 ) $$

Step 1 :

After factoring out $ -8x^{5} $ we have:

$$ -56x^{7}+32x^{6}-48x^{5} = -8x^{5} ( 7x^{2}-4x+6 ) $$

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:

$$ a = 7 , b = -4 ~ \text{ and } ~ c = 6 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 7 }$ by the constant term $\color{blue}{c = 6} $.

$$ a \cdot c = 42 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 42 $ and add to $ b = -4 $.

Step 5: All pairs of numbers with a product of $ 42 $ are:

PRODUCT = 42
1     42	-1     -42
2     21	-2     -21
3     14	-3     -14
6     7	-6     -7

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -4 }$

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ -4 }$, we conclude the polynomial cannot be factored.

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