Find the coefficient of the term indicated in the square bracke:(1 + 4𝑥)^5 using binomial expansion

(4 x + 1)^5

Expand (4 x + 1)^5 using the binomial expansion theorem.
(4 x + 1)^5 = sum_(k=0)^5 binomial(5, k) (4 x)^(5 - k) 1^k = binomial(5, 0) (4 x)^5 1^0 + binomial(5, 1) (4 x)^4 1^1 + binomial(5, 2) (4 x)^3 1^2 + binomial(5, 3) (4 x)^2 1^3 + binomial(5, 4) (4 x)^1 1^4 + binomial(5, 5) (4 x)^0 1^5:
1024 binomial(5, 0) x^5 + 256 binomial(5, 1) x^4 + 64 binomial(5, 2) x^3 + 16 binomial(5, 3) x^2 + 4 binomial(5, 4) x + binomial(5, 5)

Evaluate the binomial coefficients by looking at Pascal's triangle.
The binomial coeffients comprise the 6^th row of Pascal's triangle:
1 | | 5 | | 10 | | 10 | | 5 | | 1
(4 x)^5 + 5 (4 x)^4 + 10 (4 x)^3 + 10 (4 x)^2 + 5 4 x + 1

Distribute exponents over products in (4 x)^2.
Multiply each exponent in 4 x by 2:
(4 x)^5 + 5 (4 x)^4 + 10 (4 x)^3 + 10 4^2 x^2 + 4 5 x + 1

Evaluate 4^2.
4^2 = 16:
(4 x)^5 + 5 (4 x)^4 + 10 (4 x)^3 + 16 10 x^2 + 4 5 x + 1

Distribute exponents over products in (4 x)^3.
Multiply each exponent in 4 x by 3:
(4 x)^5 + 5 (4 x)^4 + 10 4^3 x^3 + 16 10 x^2 + 4 5 x + 1

In order to evaluate 4^3 express 4^3 as 4×4^2.
4^3 = 4×4^2:
(4 x)^5 + 5 (4 x)^4 + 4×4^2 10 x^3 + 16 10 x^2 + 4 5 x + 1

Evaluate 4^2.
4^2 = 16:
(4 x)^5 + 5 (4 x)^4 + 16 4 10 x^3 + 16 10 x^2 + 4 5 x + 1

Multiply 4 and 16 together.
4×16 = 64:
(4 x)^5 + 5 (4 x)^4 + 64 10 x^3 + 16 10 x^2 + 4 5 x + 1

Distribute exponents over products in (4 x)^4.
Multiply each exponent in 4 x by 4:
(4 x)^5 + 5 4^4 x^4 + 64 10 x^3 + 16 10 x^2 + 4 5 x + 1

Compute 4^4 by repeated squaring.
4^4 = (4^2)^2:
(4 x)^5 + (4^2)^2 5 x^4 + 64 10 x^3 + 16 10 x^2 + 4 5 x + 1

Evaluate 4^2.
4^2 = 16:
(4 x)^5 + 16^2 5 x^4 + 64 10 x^3 + 16 10 x^2 + 4 5 x + 1

(4 x)^5 + 256 5 x^4 + 64 10 x^3 + 16 10 x^2 + 4 5 x + 1

Multiply 4 and 5 together.
4×5 = 20:
(4 x)^5 + 256 5 x^4 + 64 10 x^3 + 16 10 x^2 + 20 x + 1

Multiply 16 and 10 together.
16×10 = 160:
(4 x)^5 + 256 5 x^4 + 64 10 x^3 + 160 x^2 + 20 x + 1

Multiply 64 and 10 together.
64×10 = 640:
(4 x)^5 + 256 5 x^4 + 640 x^3 + 160 x^2 + 20 x + 1

Multiply 256 and 5 together.
256×5 = 1280:
(4 x)^5 + 1280 x^4 + 640 x^3 + 160 x^2 + 20 x + 1

Distribute exponents over products in (4 x)^5.
Multiply each exponent in 4 x by 5:
4^5 x^5 + 1280 x^4 + 640 x^3 + 160 x^2 + 20 x + 1

Compute 4^5 by repeated squaring. For example a^7 = a a^6 = a (a^3)^2 = a (a a^2)^2.
4^5 = 4×4^4 = 4 (4^2)^2:
4 (4^2)^2 x^5 + 1280 x^4 + 640 x^3 + 160 x^2 + 20 x + 1

Evaluate 4^2.
4^2 = 16:
4 16^2 x^5 + 1280 x^4 + 640 x^3 + 160 x^2 + 20 x + 1

256 4 x^5 + 1280 x^4 + 640 x^3 + 160 x^2 + 20 x + 1

Multiply 4 and 256 together.
4×256 = 1024:

1024x^5 + 1280x^4 + 640x^3 + 160x^2 + 20x + 1