## Answer :

We are given the expression:

[tex]\[ x^4 + 7x^2y^2 + 16y^4 \][/tex]

To proceed, let's consider the following steps:

1.

**Identify the Components:**

- The term [tex]\( x^4 \)[/tex] is a pure fourth power of [tex]\( x \)[/tex].

- [tex]\( 7x^2y^2 \)[/tex] involves both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and is a product of squares, specifically [tex]\( (x^2)(y^2) \)[/tex], scaled by 7.

- [tex]\( 16y^4 \)[/tex] is a pure fourth power of [tex]\( y \)[/tex], scaled by 16.

2.

**Expand and Simplify:**

- Note that in this expression, each term is already fully expanded and simplified to its simplest form.

- There are no hidden factorizations or simplifications for the given terms that can further reduce the expression.

Therefore, the final expanded form of the expression [tex]\( x^4 + 7x^2y^2 + 16y^4 \)[/tex] is:

[tex]\[ \boxed{x^4 + 7x^2y^2 + 16y^4} \][/tex]

This is the simplest form we can write for this expression, and there are no further calculations or simplifications needed.