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Understanding how to square complex numbers is fundamental in complex number algebra. A complex number is represented as \(a + bi\), where \(a\) is the real component, and \(bi\) is the imaginary component, with \(i\) being the square root of -1. To square a complex number, you apply the same principles as squaring a binomial in algebra.

Following the formula \((a + bi)^{2} = a^{2} + 2abi - b^{2}i^{2}\), pay close attention to the \(i^{2}\) term. Since \(i^{2} = -1\), the term \(-b^{2}i^{2}\) simplifies to \(+b^{2}\). Remembering that the square of an imaginary unit is negative is a common point where students get tripped up. Practice with different complex numbers will help solidify this concept, ensuring it becomes second nature in your algebraic endeavors.

The next step involves adding complex numbers, which is quite straightforward once you understand the structure of a complex number. The key is to combine like terms, the real parts with real and imaginary with imaginary. In our exercise, the operation \((a^2 + 2abi - b^2) + (a^2 - 2abi - b^2)\) initially seems complex, but it’s simplified by adding the real parts \(a^2 - b^2\) and imaginary parts \(2abi - 2abi\) respectively.

Notice that the imaginary parts cancel each other out, which often happens when you add a complex number and its conjugate. This is an important property to remember because it reduces the problem to working with real numbers, making the calculations much more manageable. Students should practice with different sets to understand the pattern and gain confidence in addition of complex numbers.

The final stage of our exercise is simplifying algebraic expressions. Simplification is the process of reducing an expression into its simplest form by eliminating parentheses and combining like terms. For our problem, after adding the squared complex numbers, we're left with terms that are either strictly real or cancelled out because they are opposites.

To simplify \(2a^2 -2b^2\), notice that both terms share a common factor of 2, which we can factor out to make the expression even simpler: \(2(a^2 - b^2)\). The ability to identify common factors and use distributive properties is essential for simplifying algebraic expressions effectively. As a tip for students, always look for opportunities to combine or factor out terms in order to streamline the expression as much as possible.