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:

$$ a = 15 , b = 34 ~ \text{ and } ~ c = 15 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 15 }$ by the constant term $\color{blue}{c = 15} $.

$$ a \cdot c = 225 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 225 $ and add to $ b = 34 $.

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 $\color{blue}{ b = 34 }$

PRODUCT = 225 and SUM = 34
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 $ 34 x $ with $ 25x+9x $:

$$ 15x^{2}+34x+15 = 15x^{2}+25x+9x+15 $$

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

$$ 15x^{2}+25x+9x+15 = 5x\left(3x+5\right) + 3\left(3x+5\right) = \left(5x+3\right) \left(3x+5\right) $$

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