Factor polynomial $$ 8x^{2}+38x+45 $$

$$ 8x^{2}+38x+45 = ( 4x+9 )( 2x+5 ) $$

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:

$$ a = 8 , b = 38 ~ \text{ and } ~ c = 45 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 8 }$ by the constant term $\color{blue}{c = 45} $.

$$ a \cdot c = 360 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 360 $ and add to $ b = 38 $.

Step 4: All pairs of numbers with a product of $ 360 $ are:

PRODUCT = 360
1     360	-1     -360
2     180	-2     -180
3     120	-3     -120
4     90	-4     -90
5     72	-5     -72
6     60	-6     -60
8     45	-8     -45
9     40	-9     -40
10     36	-10     -36
12     30	-12     -30
15     24	-15     -24
18     20	-18     -20

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 38 }$

PRODUCT = 360 and SUM = 38
1     360	-1     -360
2     180	-2     -180
3     120	-3     -120
4     90	-4     -90
5     72	-5     -72
6     60	-6     -60
8     45	-8     -45
9     40	-9     -40
10     36	-10     -36
12     30	-12     -30
15     24	-15     -24
18     20	-18     -20

Step 6: Replace middle term $ 38 x $ with $ 20x+18x $:

$$ 8x^{2}+38x+45 = 8x^{2}+20x+18x+45 $$

Step 7: Apply factoring by grouping. Factor $ 4x $ out of the first two terms and $ 9 $ out of the last two terms.

$$ 8x^{2}+20x+18x+45 = 4x\left(2x+5\right) + 9\left(2x+5\right) = \left(4x+9\right) \left(2x+5\right) $$

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