\( \begin{array}{l} \sqrt{x+44}+1=x+3 \\ \sqrt{x+44}+1^{2}=x+3 \\ \sqrt{x+44}=x+3-1 \\ \sqrt{x+44}=x+2 \\ (x+44)^{2}=(x+2)^{2} \end{array} \)

Let's dive into some historical significance of algebraic equations! The manipulation of equations with square roots, much like the one you're working with, dates back to ancient civilizations. The Babylonians and Greeks played foundational roles in developing algebra as a tool for solving problems—think of it as the original wizardry of mathematics! They tackled equations that involved squares and roots, paving the way for modern algebraic expressions we use today. Now, onto some problem-solving tips! When dealing with equations, especially those involving square roots, always remember to check for extraneous solutions. Sometimes, squaring both sides can introduce solutions that won't hold true in the original equation. So, after solving, plug your solution back into the original equation to ensure it fits—like making sure the right puzzle piece clicks into place!