Factor polynomial $$ 15k^{4}+35k^{3}+20k^{2} $$

$$ 15k^{4}+35k^{3}+20k^{2} = 5k^{2}( k+1 )( 3k+4 ) $$

Step 1 :

After factoring out $ 5k^{2} $ we have:

$$ 15k^{4}+35k^{3}+20k^{2} = 5k^{2} ( 3k^{2}+7k+4 ) $$

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:

$$ a = 3 , b = 7 ~ \text{ and } ~ c = 4 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = 4} $.

$$ a \cdot c = 12 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 12 $ and add to $ b = 7 $.

Step 5: All pairs of numbers with a product of $ 12 $ are:

PRODUCT = 12
1     12	-1     -12
2     6	-2     -6
3     4	-3     -4

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 7 }$

PRODUCT = 12 and SUM = 7
1     12	-1     -12
2     6	-2     -6
3     4	-3     -4

Step 7: Replace middle term $ 7 x $ with $ 4x+3x $:

$$ 3x^{2}+7x+4 = 3x^{2}+4x+3x+4 $$

Step 8: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ 1 $ out of the last two terms.

$$ 3x^{2}+4x+3x+4 = x\left(3x+4\right) + 1\left(3x+4\right) = \left(x+1\right) \left(3x+4\right) $$

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