1. The problem asks to express the quadratic expression $x^2 + 6x + 5$ in the form $(x + a)^2 + b$, where $a$ and $b$ are constants. 2. We use the method of completing the square. The general formula is: $$x^2 + 2ax + a^2 = (x + a)^2$$ 3. Compare $x^2 + 6x + 5$ with $x^2 + 2ax + a^2$. Here, $2a = 6$ so $a = 3$. 4. Rewrite the expression: $$x^2 + 6x + 5 = (x + 3)^2 - 3^2 + 5 = (x + 3)^2 - 9 + 5 = (x + 3)^2 - 4$$ 5. Therefore, the expression in the form $(x + a)^2 + b$ is: $$(x + 3)^2 - 4$$ 6. The transformation from $y = x^2$ to $y = x^2 + 6x + 5$ can be described by rewriting the latter as: $$y = (x + 3)^2 - 4$$ 7. This means the graph of $y = x^2$ is shifted 3 units to the left (because of $x + 3$) and 4 units down (because of $-4$). 8. In terms of function notation, if $f(x) = x^2$, then the transformed function is: $$y = f(x + 3) - 4$$ Final answer: $$(x + 3)^2 - 4$$