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In mathematics, the standard form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit which satisfies the equation \(i^2 = -1\). The real part, \(a\), represents the number on the real axis, while the imaginary part, \(b\), when multiplied by \(i\), represents the number on the imaginary axis in the complex plane.

Understanding this form is crucial when performing operations such as addition, subtraction, and especially multiplication, as seen in the given exercise. Keeping the standard form in mind ensures that the real and imaginary parts are correctly identified after completing the multiplication process.

The FOIL method stands for First, Outer, Inner, Last and is used to multiply two binomials. It's a mnemonic for remembering the steps involved in the multiplication:

• First: multiply the first terms in each binomial.
• Outer: multiply the outermost terms in the product.
• Inner: multiply the innermost terms.
• Last: multiply the last terms in each binomial.

When applying the FOIL method to complex numbers, it's helpful to treat the real part and the imaginary part as the components of these binomials, being mindful of the property of the imaginary unit \(i^2 = -1\). This approach was used to solve the given exercise, distributing each part of the first complex number to the corresponding part of the second.

After applying the FOIL method, the next step in simplifying the expression involves combining like terms. Like terms are terms in an expression that have the same variable and power. In the context of complex numbers, like terms will be those terms that contain either real numbers or the imaginary unit \(i\), but not both.

For example, in the provided exercise, \( -4i \), and \( 24i \) are like terms because they are both multiples of the imaginary unit \(i\). They can be combined by adding the coefficients (the numerical parts) together to simplify the expression. Likewise, the real parts can be combined with other real parts. This is an important practice for arriving at the standard form of the result.

An imaginary number is a number that can be written as a real number multiplied by the imaginary unit \(i\), which is defined by its property \(i^2 = -1\). When we square an imaginary number, the result is a negative real number. This property is central when multiplying complex numbers.

In the context of the given exercise, \(i^2\) appears as a result of applying the FOIL method, reminding us to convert \(i^2 = -1\). Converting \(i^2\) to \(-1\) is a pivotal step in simplifying the product of complex numbers into its standard form. Recognizing imaginary numbers and their properties is essential for completing complex number multiplication accurately.