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:

$$ a = 10 , b = 51 ~ \text{ and } ~ c = 27 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 10 }$ by the constant term $\color{blue}{c = 27} $.

$$ a \cdot c = 270 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 270 $ and add to $ b = 51 $.

Step 4: All pairs of numbers with a product of $ 270 $ are:

PRODUCT = 270
1     270	-1     -270
2     135	-2     -135
3     90	-3     -90
5     54	-5     -54
6     45	-6     -45
9     30	-9     -30
10     27	-10     -27
15     18	-15     -18

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 51 }$

PRODUCT = 270 and SUM = 51
1     270	-1     -270
2     135	-2     -135
3     90	-3     -90
5     54	-5     -54
6     45	-6     -45
9     30	-9     -30
10     27	-10     -27
15     18	-15     -18

Step 6: Replace middle term $ 51 x $ with $ 45x+6x $:

$$ 10x^{2}+51x+27 = 10x^{2}+45x+6x+27 $$

Step 7: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ 3 $ out of the last two terms.

$$ 10x^{2}+45x+6x+27 = 5x\left(2x+9\right) + 3\left(2x+9\right) = \left(5x+3\right) \left(2x+9\right) $$

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