Understanding the (a + b)(c + d) Expansion
The expression (a + b)(c + d) is a fundamental algebraic formula used for multiplying two binomials. It represents the product of two sums, and its expansion is crucial for simplifying expressions and solving equations.
Expanding the Expression
The expansion of (a + b)(c + d) is based on the distributive property of multiplication. Here's how it works:

Distribute the first term of the first binomial (a) over the second binomial (c + d):
 a(c + d) = ac + ad

Distribute the second term of the first binomial (b) over the second binomial (c + d):
 b(c + d) = bc + bd

Combine the results from steps 1 and 2:
 (a + b)(c + d) = ac + ad + bc + bd
The FOIL Method
A common mnemonic for remembering this expansion is the FOIL method:
 First: Multiply the first terms of each binomial (ac).
 Outer: Multiply the outer terms of the binomials (ad).
 Inner: Multiply the inner terms of the binomials (bc).
 Last: Multiply the last terms of each binomial (bd).
This method provides a straightforward way to remember the four terms in the expanded expression.
Application in Algebra
The expansion of (a + b)(c + d) is widely used in various algebraic operations, such as:
 Simplifying complex expressions: By expanding the expression, we can combine like terms and simplify the overall expression.
 Solving quadratic equations: The expansion helps us rewrite quadratic equations in standard form, making them easier to solve.
 Factoring expressions: Knowing the expanded form can aid in factoring complex expressions by recognizing the patterns.
Examples
Here are some examples of how to expand (a + b)(c + d):
 (x + 2)(y  3):
 (x + 2)(y  3) = xy  3x + 2y  6
 (2m + 5)(3n  1):
 (2m + 5)(3n  1) = 6mn  2m + 15n  5
Conclusion
The expansion of (a + b)(c + d) is a fundamental concept in algebra with numerous applications. Understanding its expansion allows you to simplify expressions, solve equations, and factor polynomials. By applying the FOIL method or simply using the distributive property, you can efficiently expand the expression and utilize it in various algebraic manipulations.