The given expression is:

\[
n^4 + 10n^2p + 25p^2
\]

This expression resembles a quadratic form in terms of \( n^2 \). Let's analyze and factor it if possible.

### Step 1: Recognize the Form

Notice that the expression can be written as a quadratic in \( n^2 \):

\[
(n^2)^2 + 10(n^2)p + 25p^2
\]

### Step 2: Consider Factoring

We will attempt to factor this expression similarly to a quadratic in the form \( x^2 + bx + c \). Here, \( x = n^2 \), \( b = 10p \), and \( c = 25p^2 \).

The expression can be rewritten as:

\[
(n^2 + 5p)^2
\]

### Step 3: Verify the Factorization

To confirm this factorization, we expand the squared term:

\[
(n^2 + 5p)^2 = (n^2 + 5p)(n^2 + 5p)
\]

\[
= n^2 \cdot n^2 + 2 \cdot n^2 \cdot 5p + (5p)^2
\]

\[
= n^4 + 10n^2p + 25p^2
\]

The factorization is correct. Thus, the expression can be factored as:

\[
n^4 + 10n^2p + 25p^2 = (n^2 + 5p)^2
\]

### Conclusion

The given expression is a perfect square trinomial and can be expressed as:

\[
(n^2 + 5p)^2
\]

This factorization shows that the original polynomial can be represented as the square of the binomial \( n^2 + 5p \).

Would you like any further details or explanations?

### Related Questions

1. What is the importance of recognizing perfect square trinomials in algebra?
2. How can you identify and factor different types of polynomials?
3. What are the steps for factoring a quadratic expression?
4. How can you apply factorization to solve polynomial equations?
5. What are other examples of perfect square trinomials?
6. Can this factorization method be applied to polynomials of higher degrees?
7. What role does symmetry play in factorization?
8. How does the distributive property relate to polynomial factorization?

### Tip

When factoring polynomials, look for patterns such as perfect squares, differences of squares, or sums and differences of cubes to simplify the process.

Learn how to factor the expression n^4 + 10n^2p + 25p^2 into the perfect square trinomial (n^2 + 5p)^2. This problem demonstrates key concepts in algebra, including quadratic forms and perfect square trinomials. Suitable for students in Grades 10-12.

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