Expanding (a + b) * (c + d)
The expression (a + b) * (c + d) is a simple algebraic expression that represents the product of two binomials. To understand its expansion, we can use the distributive property of multiplication.
Understanding the Distributive Property
The distributive property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. In other words:
a * (b + c) = (a * b) + (a * c)
Expanding (a + b) * (c + d)
Applying the distributive property to our expression, we can expand it as follows:

Distribute (a + b) over (c + d): (a + b) * (c + d) = (a + b) * c + (a + b) * d

Distribute again: (a + b) * c + (a + b) * d = (a * c + b * c) + (a * d + b * d)

Simplify by removing the parentheses: (a * c + b * c) + (a * d + b * d) = ac + bc + ad + bd
Therefore, the expanded form of (a + b) * (c + d) is ac + bc + ad + bd.
Example
Let's consider a numerical example:
(2 + 3) * (4 + 5)
Using the expanded form, we get:
 ac + bc + ad + bd
 (2 * 4) + (3 * 4) + (2 * 5) + (3 * 5)
 8 + 12 + 10 + 15 = 45
We can verify this by directly calculating the original expression:
(2 + 3) * (4 + 5) = 5 * 9 = 45
Conclusion
Expanding (a + b) * (c + d) using the distributive property gives us a simple formula ac + bc + ad + bd. This formula is useful for simplifying algebraic expressions and solving equations.