Step 1: 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:

$$ a = 5 , b = -16 ~ \text{ and } ~ c = -45 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = -45} $.

$$ a \cdot c = -225 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -225 $ and add to $ b = -16 $.

Step 4: All pairs of numbers with a product of $ -225 $ are:

PRODUCT = -225
-1     225	1     -225
-3     75	3     -75
-5     45	5     -45
-9     25	9     -25
-15     15	15     -15

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -16 }$

PRODUCT = -225 and SUM = -16
-1     225	1     -225
-3     75	3     -75
-5     45	5     -45
-9     25	9     -25
-15     15	15     -15

Step 6: Replace middle term $ -16 x $ with $ 9x-25x $:

$$ 5x^{2}-16x-45 = 5x^{2}+9x-25x-45 $$

Step 7: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -5 $ out of the last two terms.

$$ 5x^{2}+9x-25x-45 = x\left(5x+9\right) -5\left(5x+9\right) = \left(x-5\right) \left(5x+9\right) $$

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