Step 1: Understand the goal.
The equation $\cos^4\theta + \sin^4\theta + \lambda = 0$ has real solutions only when $-\lambda$ equals some real value that $\cos^4\theta + \sin^4\theta$ can actually take. So we first find the range of that expression.
Step 2: Simplify $\cos^4\theta + \sin^4\theta$.
Use the square of the basic identity:
\[ \cos^4\theta + \sin^4\theta = (\cos^2\theta + \sin^2\theta)^2 - 2\sin^2\theta\cos^2\theta = 1 - 2\sin^2\theta\cos^2\theta \]
Step 3: Use the double angle form.
Since $\sin^2\theta\cos^2\theta = \frac14\sin^2 2\theta$, we get
\[ \cos^4\theta + \sin^4\theta = 1 - \frac12\sin^2 2\theta \]
Step 4: Find the range.
Here $\sin^2 2\theta$ runs from $0$ to $1$. So the expression runs from $1 - \frac12 = \frac12$ up to $1$.
\[ \frac12 \le \cos^4\theta + \sin^4\theta \le 1 \]
Step 5: Bring in $\lambda$.
From the equation, $\lambda = -\left(\cos^4\theta + \sin^4\theta\right)$.
Step 6: Flip the inequality with the minus sign.
Negating the bounds reverses them:
\[ -1 \le \lambda \le -\frac12 \]
\[ \boxed{\lambda \in \left[-1, -\tfrac12\right]} \]