Factor polynomial $$ 10y^{3}+31y^{2}+15y $$

$$ 10y^{3}+31y^{2}+15y = y( 5y+3 )( 2y+5 ) $$

Step 1 :

After factoring out $ y $ we have:

$$ 10y^{3}+31y^{2}+15y = y ( 10y^{2}+31y+15 ) $$

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:

$$ a = 10 , b = 31 ~ \text{ and } ~ c = 15 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 10 }$ by the constant term $\color{blue}{c = 15} $.

$$ a \cdot c = 150 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 150 $ and add to $ b = 31 $.

Step 5: All pairs of numbers with a product of $ 150 $ are:

PRODUCT = 150
1     150	-1     -150
2     75	-2     -75
3     50	-3     -50
5     30	-5     -30
6     25	-6     -25
10     15	-10     -15

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 31 }$

PRODUCT = 150 and SUM = 31
1     150	-1     -150
2     75	-2     -75
3     50	-3     -50
5     30	-5     -30
6     25	-6     -25
10     15	-10     -15

Step 7: Replace middle term $ 31 x $ with $ 25x+6x $:

$$ 10x^{2}+31x+15 = 10x^{2}+25x+6x+15 $$

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

$$ 10x^{2}+25x+6x+15 = 5x\left(2x+5\right) + 3\left(2x+5\right) = \left(5x+3\right) \left(2x+5\right) $$

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