Key idea. A composite number can be written as two integer factors greater than 1.

1. Treat the expression as a quadratic in \(n^2\). The numbers 1 and 3 have sum 4 and product 3, so

   \[n^4+4n^2+3=(n^2+1)(n^2+3).\]

2. Because \(n\) is a positive integer, \(n\ge1\). Thus \(n^2+1\ge2\) and \(n^2+3\ge4\).

3. Both factors are integers greater than 1. Their product is therefore composite for every allowed value of n.

Remember. Writing a product is not enough: check that neither factor can be 1.