Factor polynomial $$ 7x^{2}+29x+4 $$

$$ 7x^{2}+29x+4 = ( 7x+1 )( x+4 ) $$

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:

$$ a = 7 , b = 29 ~ \text{ and } ~ c = 4 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 7 }$ by the constant term $\color{blue}{c = 4} $.

$$ a \cdot c = 28 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 28 $ and add to $ b = 29 $.

Step 4: All pairs of numbers with a product of $ 28 $ are:

PRODUCT = 28
1     28	-1     -28
2     14	-2     -14
4     7	-4     -7

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 29 }$

PRODUCT = 28 and SUM = 29
1     28	-1     -28
2     14	-2     -14
4     7	-4     -7

Step 6: Replace middle term $ 29 x $ with $ 28x+x $:

$$ 7x^{2}+29x+4 = 7x^{2}+28x+x+4 $$

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

$$ 7x^{2}+28x+x+4 = 7x\left(x+4\right) + 1\left(x+4\right) = \left(7x+1\right) \left(x+4\right) $$

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