To expand \((a+5)(a+7)\), apply the distributive property (also known as FOIL for binomials):

\[
(a+5)(a+7) = a(a+7) + 5(a+7)
\]

Now distribute each term:

\[
= a^2 + 7a + 5a + 35
\]

Combine like terms:

\[
= a^2 + 12a + 35
\]

So, the expanded form of \((a+5)(a+7)\) is:

\[
a^2 + 12a + 35
\]

Would you like more details or have any questions?

Here are some related questions:
1. How do you expand other binomials like \((x+3)(x+8)\)?
2. What is the difference between factoring and expanding in algebra?
3. Can you explain the FOIL method for binomials?
4. How does expanding binomials help in solving quadratic equations?
5. What happens if the binomials include negative terms?

**Tip:** When expanding binomials, always remember to combine like terms after distributing.

Learn how to expand and simplify the binomial expression (a+5)(a+7) using the distributive property (FOIL method). This problem covers key algebraic concepts and includes step-by-step instructions to combine like terms. Suitable for students in Grades 8-10.

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