Alan

1.  3x >= 2    

3x - 2 = 4   

3x = 6

x = 2

2. 3x<2

-(3x-2) = 4

3x-2 = -4

3x = -2

x = -2/3

Sequence A:  a_n = 6^(n-1)  or  a_n = 2^(n-1)*3^(n-1)

Sequence B:  b_n = 50*3^(n-1)

For A to be larger than B:   2^(n-1)*3^(n-1)  >  50*3^(n-1)   

or   2^(n-1)  >  50  

The smallest value of n for which 2^(n-1)  >  50  is n = 7  (as 2⁵ = 32 and 2⁶ = 64) 

Hence after 7 terms A is larger than B.

Just add the f3 components in a similar manner.

Try them this way:

x-component of resultant, fx = f1*cos(theta1) + f2*cos(theta2)

y-component of rsultant,  fy = f1*sin(theta1) + f2*sin(theta2)

magnitude of resultant  f = sqrt(f1^2 + f2^2)

angle of resultant to positive x-axis  theta = tan^-1(fy/fx)

x^5 terms can only arise from the following combinations:  

2x*x^2*x^2    coeff → 2

3x^2*2x*x^2  coeff → 6

3x^2*x^2*x   coeff → 3

So overall coefficient of x^5 tterm is 2 + 6 + 3 → 11  (solution A in your list).

I should have noted that A in your list is the right answer.

Write it as (2 + 3x)^9

Expand:

(2 + 3x)^9 = 2^9 + 9*2^8*(3x) + (9*8/2!)*2^7(3x)^2 + (9*8*7/3!)*2^6*(3x)^3 + (9*8*7*6/4!)*2^5*(3x)^4 + ....

So coefficient of x^4 is (9*8*7*6/4!)*2^5*3^4 → 326592

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The analysis below should help you to work out the angles between them:

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Very nicely done Melody!