Step 1 :

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

$$ 5x^{4}+27x^{3}-18x^{2} = x^{2} ( 5x^{2}+27x-18 ) $$

Step 2 :

Step 2: 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:

$$ a = 5 , b = 27 ~ \text{ and } ~ c = -18 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = -18} $.

$$ a \cdot c = -90 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -90 $ and add to $ b = 27 $.

Step 5: All pairs of numbers with a product of $ -90 $ are:

PRODUCT = -90
-1     90	1     -90
-2     45	2     -45
-3     30	3     -30
-5     18	5     -18
-6     15	6     -15
-9     10	9     -10

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 27 }$

PRODUCT = -90 and SUM = 27
-1     90	1     -90
-2     45	2     -45
-3     30	3     -30
-5     18	5     -18
-6     15	6     -15
-9     10	9     -10

Step 7: Replace middle term $ 27 x $ with $ 30x-3x $:

$$ 5x^{2}+27x-18 = 5x^{2}+30x-3x-18 $$

Step 8: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ -3 $ out of the last two terms.

$$ 5x^{2}+30x-3x-18 = 5x\left(x+6\right) -3\left(x+6\right) = \left(5x-3\right) \left(x+6\right) $$

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