1. **State the problem:** Find the limit as $x$ approaches $-\infty$ of the function $$\frac{2x^2 + 2x^2 + 1}{x^2 + 3}.$$\n\n2. **Simplify the expression:** Combine like terms in the numerator: $$2x^2 + 2x^2 + 1 = 4x^2 + 1.$$ So the function becomes $$\frac{4x^2 + 1}{x^2 + 3}.$$\n\n3. **Identify the dominant terms:** As $x \to -\infty$, the highest power terms dominate the behavior of the function. The dominant term in the numerator is $4x^2$ and in the denominator is $x^2$.\n\n4. **Divide numerator and denominator by $x^2$ to analyze the limit:** $$\frac{4x^2 + 1}{x^2 + 3} = \frac{4 + \frac{1}{x^2}}{1 + \frac{3}{x^2}}.$$\n\n5. **Evaluate the limit:** As $x \to -\infty$, $\frac{1}{x^2} \to 0$ and $\frac{3}{x^2} \to 0$, so the expression approaches $$\frac{4 + 0}{1 + 0} = 4.$$\n\n**Final answer:** $$\lim_{x \to -\infty} \frac{2x^2 + 2x^2 + 1}{x^2 + 3} = 4.$$