https://file.aiquickdraw.com/mathbot/file/1728967353005k3ivkxuq.jpg

The given expression is:

\[
(a^2 + 16)^2 - 64a^2
\]

We need to factor this expression. Notice that this resembles the difference of squares formula, which is:

\[
x^2 - y^2 = (x - y)(x + y)
\]

Here, we can let:
\[
x = (a^2 + 16) \quad \text{and} \quad y = 8a
\]

So, we can rewrite the expression as:

\[
\left( (a^2 + 16) \right)^2 - (8a)^2
\]

Now apply the difference of squares formula:

\[
\left( (a^2 + 16) - 8a \right) \cdot \left( (a^2 + 16) + 8a \right)
\]

Thus, the expression becomes:

\[
(a^2 + 16 - 8a) \cdot (a^2 + 16 + 8a)
\]

Next, we simplify each factor:

1. \((a^2 + 16 - 8a)\) becomes \((a^2 - 8a + 16)\).
2. \((a^2 + 16 + 8a)\) becomes \((a^2 + 8a + 16)\).

Both of these are perfect square trinomials, as:

\[
a^2 - 8a + 16 = (a - 4)^2
\]
\[
a^2 + 8a + 16 = (a + 4)^2
\]

Therefore, the fully factored form of the expression is:

\[
(a - 4)^2 \cdot (a + 4)^2
\]

This matches option **C**.

Would you like further clarification or details on this solution?

### 5 Related Questions:
1. How do you recognize the difference of squares in an expression?
2. What is a perfect square trinomial and how do you factor it?
3. How can you apply the difference of squares to more complex expressions?
4. What happens if the constants in the difference of squares are not perfect squares?
5. Can factoring be used to simplify expressions in calculus problems?

### Tip:
Always check if an expression can be factored using common patterns like the difference of squares or perfect square trinomials. This can make complex problems much simpler!

Learn how to factor the algebraic expression (a^2 + 16)^2 - 64a^2 using the difference of squares method. This problem involves recognizing patterns such as perfect square trinomials and applying factoring techniques. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation