The problem asks to simplify the expression and write the result in the form $a + bi$. The given expression is:

$4 + \sqrt{-49}$

Step-by-step Solution:

1. Recognize that the square root of a negative number involves imaginary numbers. The square root of $-49$ can be rewritten as: $\sqrt{-49} = \sqrt{-1 \times 49} = \sqrt{49} \times \sqrt{-1} = 7i$ (where $i = \sqrt{-1}$).

2. Substitute this into the expression: $4 + \sqrt{-49} = 4 + 7i$

Final Answer:

$\boxed{4 + 7i}$

This is the simplified form in $a + bi$, where $a = 4$ and $b = 7$.

Would you like more details on any part of this process? Here are some related questions to explore further:

1. How do you calculate square roots of negative numbers in general?
2. What are the rules for adding and subtracting complex numbers?
3. Can you show how to multiply two complex numbers in the form $a + bi$?
4. How do you plot complex numbers on the complex plane?
5. What is the geometric interpretation of the imaginary unit $i$?

Tip: Always remember that $i^2 = -1$, which is fundamental to working with imaginary numbers.