Factor polynomial $$ -5x^{4}+70x^{3}+75x^{2} $$

$$ -5x^{4}+70x^{3}+75x^{2} = ( -5 )x^{2}( x+1 )( x-15 ) $$

Step 1 :

After factoring out $ -5x^{2} $ we have:

$$ -5x^{4}+70x^{3}+75x^{2} = -5x^{2} ( x^{2}-14x-15 ) $$

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:

$$ \color{blue}{ b = -14 } ~ \text{ and } ~ \color{red}{ c = -15 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -14 } $ and multiply to $ \color{red}{ -15 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = -15 }$.

PRODUCT = -15
-1     15	1     -15
-3     5	3     -5

Step 4: Find out which pair sums up to $\color{blue}{ b = -14 }$

PRODUCT = -15 and SUM = -14
-1     15	1     -15
-3     5	3     -5

Step 5: Put 1 and -15 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-14x-15 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-14x-15 & = (x + 1)(x -15) \end{aligned} $$

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