To rewrite the quadratic expression $x^2 + 4x + 5$ in the form $(x + a)^2 + b$, we need to complete the square.

Step 1: Completing the square

We begin by looking at the quadratic term $x^2 + 4x$. To complete the square, we need to add and subtract a constant that allows us to express $x^2 + 4x$ as a perfect square trinomial.

The coefficient of $x$ is 4, so we take half of 4, which is $2$, and then square it: $\left(\frac{4}{2}\right)^2 = 4$ Thus, we add and subtract 4 to the expression: $x^2 + 4x + 5 = (x^2 + 4x + 4) + 5 - 4$ $= (x + 2)^2 + 1$

Final Answer:

The quadratic expression $x^2 + 4x + 5$ can be written as: $(x + 2)^2 + 1$ In this case, $a = 2$ and $b = 1$.

Would you like any further details or explanations?

Here are 5 follow-up questions related to completing the square:

1. What happens if the coefficient of $x^2$ is not 1? How would you complete the square in that case?
2. How can completing the square help solve quadratic equations?
3. Can all quadratic expressions be written in the form $(x + a)^2 + b$?
4. How does completing the square relate to the quadratic formula?
5. How would this method apply to a quadratic expression with a negative constant term?

Tip: Completing the square is particularly useful when converting a quadratic equation into vertex form, which helps in identifying the vertex of the parabola represented by the quadratic function.