1. **Problem statement:** Find the least squares approximating polynomial of degree 2 for the function $f(x)=x^2-2x+3$. 2. **Understanding the problem:** The least squares approximating polynomial of degree 2 is a polynomial $p(x)=a_0 + a_1 x + a_2 x^2$ that best fits the function $f(x)$ in the least squares sense over a given interval or set of points. 3. **Key formula:** The coefficients $a_0, a_1, a_2$ minimize the integral of the squared difference: $$\int (f(x) - p(x))^2 \, dx$$ 4. **Since $f(x)$ is already a polynomial of degree 2:** $$f(x) = x^2 - 2x + 3$$ 5. **The best least squares polynomial of degree 2 that approximates $f(x)$ is $f(x)$ itself, because it is already degree 2.** 6. **Therefore, the least squares approximating polynomial is:** $$p(x) = 1 \cdot x^2 - 2 \cdot x + 3$$ 7. **Final answer:** $$\boxed{p(x) = x^2 - 2x + 3}$$