Use Pascal's Δ           1 
                            1     1
                         1     2     1
                      1    3     3     1
                    1    4    6     4     1
                 1    5   10   10   5     1 
              1     6   15   20   15   6    1
(a + b)⁶ = a⁶ + 6a⁵b + 15a⁴b² + 20a³b³ + 15a²b⁴ + 6ab⁵ + b⁶
1. If you notice, that coefficients are from the bottom of Pascal 's triangle.
2. The powers of a go down, from 6 to 0;
3. The exponents of b go up, from 0 to 6;
4. The sum of the powers of each therm equal 6;
5. If you multiply all coefficients you will get 2⁶ = 64 (for each row of Pascal's Δ 2ⁿ)
Now replace a by 7 and b by (-4x)
7⁶ - 6 · 7⁵ · 4x + 15 · 7⁴ · (-4x)² - 20 · 7³ · (4x)³ + 15 · 7² · (-4x)⁴ - 6 · 7 · (4x)⁵ + (-4x)⁶

(7 – 4x)^6

= 7^6 + (6*7^5)(-4x) + (15*7^4)(-4x)^2 + (20*7^3)(-4x)^3 + (15*7^2)(-4x)^4 + (6*7) (-4x)^5 + (-4x)^6

= 7^6 - (6*7^5)(4x) + (15*7^4)(16x^2) - (20*7^3)(64 x^3) + (15*7^2)(256 x^4) - (6*7)(1024 x^5) + 4096

The rest is arithmetic.

You can use binomial expansion.

(7-4x)^6 = ∑{i = 0, 6} C(6, i) (7)^(6-i) (-4x)^i

You can expand binomials using Pascal's Triangle to find the exponents

      1
     1 1
    1 2 1
   1 3 3 1
  1 4 6 4 1
1 5 10 10 5 1 n=6

This can be formed by starting with 1, then row 2 is 1,1, then the two ones in row two add to make the 2 in row 3, etc. 

rewrite the binomial as (-4x+7)6 for simplicity's sake

We split the binomial into 2 terms and the exponent for x decreases while the other exponent increases, then multiply by the appropriate number in Pascal's Triangle.

1(-4x)⁶(7)⁰= -4096x⁶

5(-4x)⁵(7)¹= -35840x⁵

etc...