To expand the expression \((x+5)(x+7)\), you can use the distributive property (also known as FOIL for binomials):

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

1. Distribute \(x\) to \((x+7)\):
\[
x(x+7) = x^2 + 7x
\]

2. Distribute \(5\) to \((x+7)\):
\[
5(x+7) = 5x + 35
\]

3. Combine the terms:
\[
x^2 + 7x + 5x + 35 = x^2 + 12x + 35
\]

The expanded form is:

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

Would you like me to break this down further or explore related concepts? 

### Questions for further exploration:
1. What happens when you substitute values into the expanded form versus the factored form?
2. How can this expansion be reversed through factoring?
3. How does this relate to completing the square for a quadratic equation?
4. Can we graph this quadratic equation to visualize its behavior?
5. What are the roots of the equation \((x+5)(x+7) = 0\)?

### Tip:
When expanding or factoring polynomials, always double-check your combination of like terms to ensure accuracy!

Explore how to expand and simplify the binomial expression (x+5)(x+7) step-by-step. Learn key algebraic concepts such as the distributive property and combining like terms, resulting in the quadratic equation x^2 + 12x + 35. Suitable for students in Grades 8-10.

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