Factor polynomial $$ 7k^{2}+38k+40 $$

$$ 7k^{2}+38k+40 = ( 7k+10 )( k+4 ) $$

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:

$$ a = 7 , b = 38 ~ \text{ and } ~ c = 40 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 7 }$ by the constant term $\color{blue}{c = 40} $.

$$ a \cdot c = 280 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 280 $ and add to $ b = 38 $.

Step 4: All pairs of numbers with a product of $ 280 $ are:

PRODUCT = 280
1     280	-1     -280
2     140	-2     -140
4     70	-4     -70
5     56	-5     -56
7     40	-7     -40
8     35	-8     -35
10     28	-10     -28
14     20	-14     -20

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 38 }$

PRODUCT = 280 and SUM = 38
1     280	-1     -280
2     140	-2     -140
4     70	-4     -70
5     56	-5     -56
7     40	-7     -40
8     35	-8     -35
10     28	-10     -28
14     20	-14     -20

Step 6: Replace middle term $ 38 x $ with $ 28x+10x $:

$$ 7x^{2}+38x+40 = 7x^{2}+28x+10x+40 $$

Step 7: Apply factoring by grouping. Factor $ 7x $ out of the first two terms and $ 10 $ out of the last two terms.

$$ 7x^{2}+28x+10x+40 = 7x\left(x+4\right) + 10\left(x+4\right) = \left(7x+10\right) \left(x+4\right) $$

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