Factor polynomial $$ 4x^{2}+59x+195 $$

$$ 4x^{2}+59x+195 = ( x+5 )( 4x+39 ) $$

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:

$$ a = 4 , b = 59 ~ \text{ and } ~ c = 195 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = 195} $.

$$ a \cdot c = 780 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 780 $ and add to $ b = 59 $.

Step 4: All pairs of numbers with a product of $ 780 $ are:

PRODUCT = 780
1     780	-1     -780
2     390	-2     -390
3     260	-3     -260
4     195	-4     -195
5     156	-5     -156
6     130	-6     -130
10     78	-10     -78
12     65	-12     -65
13     60	-13     -60
15     52	-15     -52
20     39	-20     -39
26     30	-26     -30

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 59 }$

PRODUCT = 780 and SUM = 59
1     780	-1     -780
2     390	-2     -390
3     260	-3     -260
4     195	-4     -195
5     156	-5     -156
6     130	-6     -130
10     78	-10     -78
12     65	-12     -65
13     60	-13     -60
15     52	-15     -52
20     39	-20     -39
26     30	-26     -30

Step 6: Replace middle term $ 59 x $ with $ 39x+20x $:

$$ 4x^{2}+59x+195 = 4x^{2}+39x+20x+195 $$

Step 7: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ 5 $ out of the last two terms.

$$ 4x^{2}+39x+20x+195 = x\left(4x+39\right) + 5\left(4x+39\right) = \left(x+5\right) \left(4x+39\right) $$

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