To simplify the rational expression \( \frac{k^2}{2k^2 + 4k + 5} \), we first look for factors in the denominator.
1. **Factor the denominator**: The quadratic expression in the denominator is \( 2k^2 + 4k + 5 \). We can check if it can be factored:
- The discriminant of the quadratic \( b^2 – 4ac \) is \( 4^2 – 4 \cdot 2 5 = 16 – 40 = -24 \), which is negative.
- This means the quadratic does not factor over the real numbers and remains as it is.
2. **Rewrite the expression**: Since the denominator does not factor, we just rewrite the rational expression as it is:
\( \frac{k^2}{2k^2 + 4k + 5} \)
3. **Identify restrictions**: The restrictions come from the denominator. We need to find the values of \( k \) that make the denominator zero:
- Set the denominator equal to zero: \( 2k^2 + 4k + 5 = 0 \).
- Since the discriminant is negative, there are no real values of \( k \) that make the denominator zero.
- Thus, the expression is defined for all real numbers.
In conclusion, the simplified form of the rational expression is:
\( \frac{k^2}{2k^2 + 4k + 5} \)
And the restrictions on the variable \( k \) are that there are no restrictions; it is defined for all real values of \( k \).