Step 1 :

After factoring out $ x^{2} $ we have:

$$ 2x^{4}+21x^{3}+49x^{2} = x^{2} ( 2x^{2}+21x+49 ) $$

Step 2 :

Step 2: 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:

$$ a = 2 , b = 21 ~ \text{ and } ~ c = 49 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = 49} $.

$$ a \cdot c = 98 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 98 $ and add to $ b = 21 $.

Step 5: All pairs of numbers with a product of $ 98 $ are:

PRODUCT = 98
1     98	-1     -98
2     49	-2     -49
7     14	-7     -14

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 21 }$

PRODUCT = 98 and SUM = 21
1     98	-1     -98
2     49	-2     -49
7     14	-7     -14

Step 7: Replace middle term $ 21 x $ with $ 14x+7x $:

$$ 2x^{2}+21x+49 = 2x^{2}+14x+7x+49 $$

Step 8: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ 7 $ out of the last two terms.

$$ 2x^{2}+14x+7x+49 = 2x\left(x+7\right) + 7\left(x+7\right) = \left(2x+7\right) \left(x+7\right) $$

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