Step 1 :

After factoring out $ 3a $ we have:

$$ 1029a^{4}+648a = 3a ( 343a^{3}+216 ) $$

Step 2 :

To factor $ 343a^{3}+216 $ we can use sum of cubes formula:

$$ I^3 - II^3 = (I + II)(I^2 - I \cdot II + II^2) $$

After putting $ I = 7a $ and $ II = 6 $ , we have:

$$ 343a^{3}+216 = ( 7a+6 ) ( 49a^{2}-42a+36 ) $$

Step 3 :

Step 3: 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:

$$ a = 49 , b = -42 ~ \text{ and } ~ c = 36 $$

Step 4: Multiply the leading coefficient $\color{blue}{ a = 49 }$ by the constant term $\color{blue}{c = 36} $.

$$ a \cdot c = 1764 $$

Step 5: Find out two numbers that multiply to $ a \cdot c = 1764 $ and add to $ b = -42 $.

Step 6: All pairs of numbers with a product of $ 1764 $ are:

PRODUCT = 1764
1     1764	-1     -1764
2     882	-2     -882
3     588	-3     -588
4     441	-4     -441
6     294	-6     -294
7     252	-7     -252
9     196	-9     -196
12     147	-12     -147
14     126	-14     -126
18     98	-18     -98
21     84	-21     -84
28     63	-28     -63
36     49	-36     -49
42     42	-42     -42

Step 7: Find out which factor pair sums up to $\color{blue}{ b = -42 }$

Step 8: Because none of these pairs will give us a sum of $ \color{blue}{ -42 }$, we conclude the polynomial cannot be factored.

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