Factor polynomial $$ 24x^{2}+70x+49 $$

$$ 24x^{2}+70x+49 = ( 6x+7 )( 4x+7 ) $$

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:

$$ a = 24 , b = 70 ~ \text{ and } ~ c = 49 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 24 }$ by the constant term $\color{blue}{c = 49} $.

$$ a \cdot c = 1176 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 1176 $ and add to $ b = 70 $.

Step 4: All pairs of numbers with a product of $ 1176 $ are:

PRODUCT = 1176
1     1176	-1     -1176
2     588	-2     -588
3     392	-3     -392
4     294	-4     -294
6     196	-6     -196
7     168	-7     -168
8     147	-8     -147
12     98	-12     -98
14     84	-14     -84
21     56	-21     -56
24     49	-24     -49
28     42	-28     -42

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 70 }$

PRODUCT = 1176 and SUM = 70
1     1176	-1     -1176
2     588	-2     -588
3     392	-3     -392
4     294	-4     -294
6     196	-6     -196
7     168	-7     -168
8     147	-8     -147
12     98	-12     -98
14     84	-14     -84
21     56	-21     -56
24     49	-24     -49
28     42	-28     -42

Step 6: Replace middle term $ 70 x $ with $ 42x+28x $:

$$ 24x^{2}+70x+49 = 24x^{2}+42x+28x+49 $$

Step 7: Apply factoring by grouping. Factor $ 6x $ out of the first two terms and $ 7 $ out of the last two terms.

$$ 24x^{2}+42x+28x+49 = 6x\left(4x+7\right) + 7\left(4x+7\right) = \left(6x+7\right) \left(4x+7\right) $$

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