The question in Problem 26 asks:

"What number must be added to $x^2 + 4x + 1$ to form a perfect square?"

To form a perfect square trinomial, we can use the process of completing the square. Here's the step-by-step method:

Step 1: Identify the coefficient of the linear term.

The given quadratic is $x^2 + 4x + 1$. The coefficient of the linear term $4x$ is $4$.

Step 2: Divide the linear coefficient by 2 and square it.

Take half of the linear coefficient $4$, which is: $\frac{4}{2} = 2$ Now, square it: $2^2 = 4$

Step 3: Adjust the constant term.

To complete the square, the constant term should be $4$, but the current constant term is $1$.

Thus, the number that must be added is: $4 - 1 = 3$

Final Answer:

The number that must be added is 3 (Option D).

Would you like more details or explanations?

Here are 5 related questions to explore:

1. What is the process of completing the square for different types of quadratics?
2. How do you identify perfect square trinomials quickly?
3. What happens if the coefficient of $x^2$ is not 1 when completing the square?
4. Can completing the square be used to solve quadratic equations?
5. How does completing the square relate to the quadratic formula?

Tip: Completing the square is especially useful for converting quadratic equations into vertex form, which makes graphing easier!