Expanding (x+1)^4
The expansion of (x+1)^4 can be done using the binomial theorem or by repeated multiplication.
Expanding using the Binomial Theorem
The binomial theorem states that:
(x + y)^n = ∑ (n choose k) * x^(nk) * y^k
where:
 n is a nonnegative integer (the power of the binomial)
 k is an integer that ranges from 0 to n
 (n choose k) is the binomial coefficient, calculated as n! / (k! * (nk)!)
Applying this to (x+1)^4:
 n = 4
 k goes from 0 to 4
Therefore, the expansion is:
(x + 1)^4 = (4 choose 0) * x^4 * 1^0 + (4 choose 1) * x^3 * 1^1 + (4 choose 2) * x^2 * 1^2 + (4 choose 3) * x^1 * 1^3 + (4 choose 4) * x^0 * 1^4
Calculating the binomial coefficients:
 (4 choose 0) = 1
 (4 choose 1) = 4
 (4 choose 2) = 6
 (4 choose 3) = 4
 (4 choose 4) = 1
Substituting the coefficients:
(x + 1)^4 = 1 * x^4 + 4 * x^3 + 6 * x^2 + 4 * x + 1
Therefore, (x + 1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1
Expanding by Repeated Multiplication
We can also expand (x+1)^4 by repeatedly multiplying:
(x + 1)^4 = (x + 1) * (x + 1) * (x + 1) * (x + 1)

Multiply the first two factors:
(x + 1) * (x + 1) = x^2 + 2x + 1

Multiply the result by the third factor:
(x^2 + 2x + 1) * (x + 1) = x^3 + 3x^2 + 3x + 1

Multiply the result by the fourth factor:
(x^3 + 3x^2 + 3x + 1) * (x + 1) = x^4 + 4x^3 + 6x^2 + 4x + 1
Therefore, (x + 1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1
Conclusion
Both methods lead to the same result: (x + 1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1. The binomial theorem is a more efficient method for higher powers, while repeated multiplication is simpler for smaller powers.