Factor polynomial $$ -x^{5}-7x^{4}+49x^{3}+343x^{2} $$

$$ -x^{5}-7x^{4}+49x^{3}+343x^{2} = -x^{2}( x-7 )( x+7 )^{ 2 } $$

Step 1 :

After factoring out $ -x^{2} $ we have:

$$ -x^{5}-7x^{4}+49x^{3}+343x^{2} = -x^{2} ( x^{3}+7x^{2}-49x-343 ) $$

Step 2 :

To factor $ x^{3}+7x^{2}-49x-343 $ we can use factoring by grouping:

Group $ \color{blue}{ x^{3} }$ with $ \color{blue}{ 7x^{2} }$ and $ \color{red}{ -49x }$ with $ \color{red}{ -343 }$ then factor each group.

$$ \begin{aligned} x^{3}+7x^{2}-49x-343 = ( \color{blue}{ x^{3}+7x^{2} } ) + ( \color{red}{ -49x-343 }) &= \\ &= \color{blue}{ x^{2}( x+7 )} + \color{red}{ -49( x+7 ) } = \\ &= (x^{2}-49)(x+7) \end{aligned} $$

Step 3 :

Rewrite $ x^{2}-49 $ as:

$$ x^{2}-49 = (x)^2 - (7)^2 $$

Now we can apply the difference of squares formula.

$$ I^2 - II^2 = (I - II)(I + II) $$

After putting $ I = x $ and $ II = 7 $ , we have:

$$ x^{2}-49 = (x)^2 - (7)^2 = ( x-7 ) ( x+7 ) $$

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