Expanding the Expression (a + b + c)(d + e + f)
The expression (a + b + c)(d + e + f) represents the product of two trinomials. To expand this expression, we can use the distributive property.
The Distributive Property
The distributive property states that for any real numbers a, b, and c:
a(b + c) = ab + ac
Expanding the Expression
We can apply the distributive property twice to expand (a + b + c)(d + e + f):

Distribute the first trinomial (a + b + c) over the second trinomial (d + e + f): (a + b + c)(d + e + f) = a(d + e + f) + b(d + e + f) + c(d + e + f)

Distribute each term of the first trinomial over the second trinomial: ad + ae + af + bd + be + bf + cd + ce + cf
The Result
Therefore, the expanded form of (a + b + c)(d + e + f) is:
ad + ae + af + bd + be + bf + cd + ce + cf
Key Points
 The expanded form contains nine terms, each representing the product of one term from the first trinomial and one term from the second trinomial.
 This expansion demonstrates the power of the distributive property in simplifying and manipulating algebraic expressions.
Example
Let's consider an example:
If a = 2, b = 3, c = 1, d = 4, e = 5, and f = 6, then:
(a + b + c)(d + e + f) = (2 + 3 + 1)(4 + 5 + 6) = 6 * 15 = 90
We can verify this by substituting the values into the expanded form:
24 + 25 + 26 + 34 + 35 + 36 + 14 + 15 + 1*6 = 8 + 10 + 12 + 12 + 15 + 18 + 4 + 5 + 6 = 90