Step  1  :

343 SIMPLIFY ——— 729

Equation at the end of step  1  :

343 ((p3) • (q3)) + ——— 729

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a fraction to a whole 

Rewrite the whole as a fraction using  729  as the DENOMINATOR :

p3q3 p3q3 • 729 p3q3 = ———— = —————————— 1 729

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole 

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 2.2       Adding up the two equivalent fractions 
Add the two equivalent fractions which now have a common denominator

Combine the NUMERATORS together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

p3q3 • 729 + 343 729p3q3 + 343 ———————————————— = ————————————— 729 729

Trying to factor as a Sum of Cubes :

 2.3      Factoring:  729p3q3 + 343 

Theory : A sum of two perfect cubes,  a3 + b3can be factored into  :
             (a+b) • (a2-ab+b2)
Proof  : (a+b) • (a2-ab+b2) = 
    a3-a2b+ab2+ba2-b2a+b3 =
    a3+(a2b-ba2)+(ab2-b2a)+b3=
    a3+0+0+b3=
    a3+b3

Check :  729  is the cube of  9 

Check :  343  is the cube of   7 
Check :  p3 is the cube of   p1

Check :  q3 is the cube of   q1

Factorization is :
             (9pq + 7)  •  (81p2q2 - 63pq + 49) 

Trying to factor a MULTI variable polynomial :

 2.4    Factoring    81p2q2 - 63pq + 49 

Try to factor this multi-variable trinomial using trial and error 

 Factorization fails

Final result :

(9pq + 7) • (81p2q2 - 63pq + 49) ———————————————————————————————— 729