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The difference of squares is a handy formula when dealing with certain binomials. This formula states that for any two numbers or variables, a and b, the expression \((a + b)(a - b)\) can be simplified to \(a^2 - b^2\). It is particularly useful when one of the terms is composed of complex numbers. In our exercise \((4 + 9i)(4 - 9i)\), we can see that it perfectly fits the difference of squares pattern.By applying the formula \(a^2 - b^2\), we convert our original complex binomials into a simpler form. Here, \(a = 4\) and \(b = 9i\). Substituting these values into \(a^2 - b^2\), we get \(4^2 - (9i)^2\). This step is powerful because it reduces what could be a complex multiplication into straightforward arithmetic.

The imaginary unit, denoted by \i\, is fundamental when working with complex numbers. The defining property of \(i\) is that \(i^2 = -1\). Understanding this property is key to simplifying expressions involving complex numbers.In our specific exercise, after applying the difference of squares, an important step is simplifying \(-(9i)^2\). Knowing that \(i^2 = -1\), we substitute as follows: \-(81i^2) = -(81(-1)) = 81\. This conversion allows us to turn what initially seems like a complex multiplication into a simple addition.

Binomials are expressions that contain two terms. In the context of complex numbers, binomials often take the form \a + bi\, where \a\ and \b\ are real numbers and \(i\) is the imaginary unit. For our exercise, the binomials are \4 + 9i\ and \4 - 9i\.To multiply these binomials, we can use the difference of squares formula because one binomial is the conjugate of the other. This approach simplifies the multiplication significantly. Remember that conjugate pairs, like our binomials, always result in an expression without the imaginary unit in the final answer.