Expanding (a + b + c + d)^2
The expression (a + b + c + d)^2 represents the square of the sum of four variables. Expanding this expression involves applying the distributive property multiple times.
Understanding the Formula
The formula for expanding (a + b + c + d)^2 is:
(a + b + c + d)^2 = a^2 + b^2 + c^2 + d^2 + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd
Here's a breakdown of the formula:
 Squares of individual terms: The first four terms (a^2, b^2, c^2, d^2) represent the squares of each individual variable.
 Crossproducts: The remaining terms (2ab, 2ac, 2ad, 2bc, 2bd, 2cd) represent the products of each pair of variables, multiplied by 2.
Derivation of the Formula
The formula can be derived using the distributive property:

(a + b + c + d)^2 = (a + b + c + d) * (a + b + c + d)

Expand the expression by distributing each term:
 a * (a + b + c + d)
 b * (a + b + c + d)
 c * (a + b + c + d)
 d * (a + b + c + d)

Simplify by multiplying each term:
 a^2 + ab + ac + ad
 ab + b^2 + bc + bd
 ac + bc + c^2 + cd
 ad + bd + cd + d^2

Combine like terms:
 a^2 + b^2 + c^2 + d^2 + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd
Therefore, we arrive at the formula (a + b + c + d)^2 = a^2 + b^2 + c^2 + d^2 + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd
Applications
This formula has various applications in algebra, calculus, and other areas of mathematics. It's particularly useful for:
 Simplifying expressions: It allows you to expand and simplify complex expressions involving squares of sums.
 Solving equations: It can be used to manipulate equations and find solutions.
 Deriving other formulas: It serves as a basis for deriving other formulas in algebra and calculus.
Conclusion
The formula (a + b + c + d)^2 = a^2 + b^2 + c^2 + d^2 + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd provides a systematic way to expand the square of a sum of four variables. This formula has wide applications in mathematics and is a fundamental tool for simplifying expressions and solving equations.