To factor the expression \( 5n^5 - n^4 + 5n^3 - n^2 \), we can first group the terms in a way that makes factoring easier. Let's rearrange the expression:
\[ 5n^5 + 5n^3 - n^4 - n^2. \]
Next, we can group the terms:
\[ (5n^5 + 5n^3) + (-n^4 - n^2). \]
Now we can factor out common factors from each group:
- From the first group \( 5n^5 + 5n^3 \), we can factor out \( 5n^3 \):
\[ 5n^3(n^2 + 1). \]
- From the second group \( -n^4 - n^2 \), we can factor out \(-n^2\):
\[ -n^2(n^2 + 1). \]
Now, our expression becomes:
\[ 5n^3(n^2 + 1) - n^2(n^2 + 1). \]
Notice that we have a common factor of \((n^2 + 1)\):
\[ (5n^3 - n^2)(n^2 + 1). \]
Next, we can factor out \(n^2\) from the first part:
\[ n^2(5n - 1)(n^2 + 1). \]
Putting it all together, the completely factored form of the expression \( 5n^5 - n^4 + 5n^3 - n^2 \) is:
\[ n^2(5n - 1)(n^2 + 1). \]