Step 1 :

After factoring out $ -2x $ we have:

$$ -2x^{3}-8x^{2}+90x = -2x ( x^{2}+4x-45 ) $$

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:

$$ \color{blue}{ b = 4 } ~ \text{ and } ~ \color{red}{ c = -45 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 4 } $ and multiply to $ \color{red}{ -45 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = -45 }$.

PRODUCT = -45
-1     45	1     -45
-3     15	3     -15
-5     9	5     -9

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

PRODUCT = -45 and SUM = 4
-1     45	1     -45
-3     15	3     -15
-5     9	5     -9

Step 5: Put -5 and 9 into placeholders to get factored form.

$$ \begin{aligned} x^{2}+4x-45 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+4x-45 & = (x -5)(x + 9) \end{aligned} $$

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