Factor polynomial $$ 4m^{2}+15m+9 $$

$$ 4m^{2}+15m+9 = ( 4m+3 )( m+3 ) $$

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:

$$ a = 4 , b = 15 ~ \text{ and } ~ c = 9 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = 9} $.

$$ a \cdot c = 36 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 36 $ and add to $ b = 15 $.

Step 4: All pairs of numbers with a product of $ 36 $ are:

PRODUCT = 36
1     36	-1     -36
2     18	-2     -18
3     12	-3     -12
4     9	-4     -9
6     6	-6     -6

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 15 }$

PRODUCT = 36 and SUM = 15
1     36	-1     -36
2     18	-2     -18
3     12	-3     -12
4     9	-4     -9
6     6	-6     -6

Step 6: Replace middle term $ 15 x $ with $ 12x+3x $:

$$ 4x^{2}+15x+9 = 4x^{2}+12x+3x+9 $$

Step 7: Apply factoring by grouping. Factor $ 4x $ out of the first two terms and $ 3 $ out of the last two terms.

$$ 4x^{2}+12x+3x+9 = 4x\left(x+3\right) + 3\left(x+3\right) = \left(4x+3\right) \left(x+3\right) $$

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