Factor polynomial $$ -2x^{4}-34x^{3}-140x^{2} $$

$$ -2x^{4}-34x^{3}-140x^{2} = ( -2 )x^{2}( x+7 )( x+10 ) $$

Step 1 :

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

$$ -2x^{4}-34x^{3}-140x^{2} = -2x^{2} ( x^{2}+17x+70 ) $$

Step 2 :

Step 2: Identify constants $ \color{blue}{ b }$ and $\color{red}{ c }$. ( $ \color{blue}{ b }$ is a number in front of the $ x $ term and $ \color{red}{ c } $ is a constant). In our case:

$$ \color{blue}{ b = 17 } ~ \text{ and } ~ \color{red}{ c = 70 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 17 } $ and multiply to $ \color{red}{ 70 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 70 }$.

PRODUCT = 70
1     70	-1     -70
2     35	-2     -35
5     14	-5     -14
7     10	-7     -10

Step 4: Find out which pair sums up to $\color{blue}{ b = 17 }$

PRODUCT = 70 and SUM = 17
1     70	-1     -70
2     35	-2     -35
5     14	-5     -14
7     10	-7     -10

Step 5: Put 7 and 10 into placeholders to get factored form.

$$ \begin{aligned} x^{2}+17x+70 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+17x+70 & = (x + 7)(x + 10) \end{aligned} $$

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