1. **State the problem:** We start with the function $f(x) = x^2$ and apply transformations to get $g(x) = x^2 + 4x + 5$. We want to understand how $g(x)$ relates to $f(x)$ and confirm the transformations. 2. **Recall the transformations:** Shifting $f(x) = x^2$ 2 units left means replacing $x$ by $x + 2$, so the function becomes $f(x+2) = (x+2)^2$. 3. **Shift 1 unit up:** After shifting left, shifting up by 1 unit means adding 1 to the function: $f(x+2) + 1 = (x+2)^2 + 1$. 4. **Expand and simplify:** $$ (x+2)^2 + 1 = (x^2 + 4x + 4) + 1 = x^2 + 4x + 5 $$ 5. **Compare with $g(x)$:** The expression matches $g(x) = x^2 + 4x + 5$, confirming that $g$ is the graph of $f$ shifted 2 units left and 1 unit up. 6. **Summary:** The graph of $g$ is the graph of $f(x) = x^2$ shifted 2 units left and 1 unit up, exactly as described. **Final answer:** $g(x) = (x+2)^2 + 1$ represents the graph of $f(x) = x^2$ shifted 2 units left and 1 unit up.