How do you simplify the rational expression 10n^2 / (24n^4 + 9n^2 + 18) and what are the restrictions on the variable?

To simplify the rational expression 10n² / (24n⁴ + 9n² + 18), we first need to factor the denominator.

The expression in the denominator is 24n⁴ + 9n² + 18. We can factor this expression by looking for common factors and simplifying it step-by-step:

• First, we can factor out a common factor of 3:
  3(8n⁴ + 3n² + 6).
• Next, we need to see if the quadratic expression (8n⁴ + 3n² + 6) can be factored further. However, this part does not factor nicely into real numbers.

Now the expression looks like this:

10n² / (3(8n⁴ + 3n² + 6)).

This is the simplified form of our expression. However, we should also consider the restrictions on the variable n.

For the original rational expression to be defined, the denominator cannot be zero. Thus, we need to solve:

24n⁴ + 9n² + 18 = 0.

Since the discriminant (b² – 4ac) of this particular polynomial does not yield real roots, we find that the denominator is never zero for real values of n. Hence, the only restriction is that n cannot be undefined.

This leads us to conclude that the rational expression is valid except for n being equal to any non-real solutions.