Simplifying (a+b)(c+d)
The expression (a+b)(c+d) is a product of two binomials. To simplify it, we can use the distributive property of multiplication.
Here's how it works:

Expand the first binomial: Think of (a+b) as a single entity and multiply it by each term inside the second binomial (c+d).
(a+b)(c+d) = (a+b) * c + (a+b) * d

Apply the distributive property again: Now distribute the 'c' and 'd' to the terms inside the parentheses.
(a+b) * c + (a+b) * d = ac + bc + ad + bd
Therefore, the simplified form of (a+b)(c+d) is ac + bc + ad + bd.
Understanding the Concept
This method is essentially a visual representation of the FOIL method, which stands for First, Outer, Inner, Last.
 First: a * c
 Outer: a * d
 Inner: b * c
 Last: b * d
Remember, FOIL is simply a mnemonic device for remembering the steps of the distributive property when multiplying two binomials.
Example
Let's say we have the expression (x + 2)(y + 3). Using the distributive property (or FOIL):
 First: x * y = xy
 Outer: x * 3 = 3x
 Inner: 2 * y = 2y
 Last: 2 * 3 = 6
Combining the terms, we get: xy + 3x + 2y + 6
Conclusion
Simplifying (a+b)(c+d) using the distributive property is a fundamental concept in algebra. Understanding this process is crucial for working with polynomials and solving algebraic equations.