Factor polynomial $$ 2y^{2}+27y+13 $$

$$ 2y^{2}+27y+13 = ( 2y+1 )( y+13 ) $$

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:

$$ a = 2 , b = 27 ~ \text{ and } ~ c = 13 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = 13} $.

$$ a \cdot c = 26 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 26 $ and add to $ b = 27 $.

Step 4: All pairs of numbers with a product of $ 26 $ are:

PRODUCT = 26
1     26	-1     -26
2     13	-2     -13

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 27 }$

PRODUCT = 26 and SUM = 27
1     26	-1     -26
2     13	-2     -13

Step 6: Replace middle term $ 27 x $ with $ 26x+x $:

$$ 2x^{2}+27x+13 = 2x^{2}+26x+x+13 $$

Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ 1 $ out of the last two terms.

$$ 2x^{2}+26x+x+13 = 2x\left(x+13\right) + 1\left(x+13\right) = \left(2x+1\right) \left(x+13\right) $$

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