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When adding complex numbers, we combine the real parts with real parts and the imaginary parts with imaginary parts. A complex number is in the form of \(a + bi\), where \(a\) represents the real part and \(bi\) represents the imaginary part with \(i\) being the imaginary unit.

To add complex numbers, simply add the real components as one group and the imaginary components as another. For instance, if we have two complex numbers such as \(3 + 4i\) and \(2 + 5i\), their sum would be \((3 + 2) + (4i + 5i) = 5 + 9i\). This operation is akin to adding two vectors in a plane, where each component is treated independently.

Subtracting complex numbers involves a similar process to addition, but we subtract the real and imaginary parts separately. If we take the example \(3 + 4i\) and \(2 + 5i\) again, but this time we want to subtract the second from the first, it looks like \((3 - 2) + (4i - 5i) = 1 - i\).

It's important to pay attention to the signs. When a complex number is preceded by a negative sign, as seen in our original exercise, both the real and imaginary parts will switch signs when the negative is distributed across the number. This is a common place where mistakes can occur, so ensure the distribution of negative signs correctly adjusts both parts of the complex number.

In mathematics, the order in which we execute operations is critical to arriving at the correct answer. This order can be remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

This rule applies when working with complex numbers as well. In the given exercise, the parentheses indicate the numbers that are treated as a unit, and when removing them, we need to take care of the sign in front of each. Additionally, any exponents associated with \(i\), which can appear in more complex problems, should be dealt with according to the exponent rules for \(i\).

The concept of combining like terms is essential for simplifying expressions, whether they're real or complex numbers. Like terms have the same variable parts, that is, they can be both real numbers or multiples of \(i\), the imaginary unit.

When faced with an expression like \(7 + 9 + 2i + 17 + i\), we combine the like terms to find a simpler form. This means we add up the real numbers \(7 + 9 + 17\), which gives us 33, and the imaginary numbers \(2i + i\), which give us \(3i\). Therefore, the expression simplifies to \(33 + 3i\), which is in the standard form for a complex number: a real part and an imaginary part. With practice, combining like terms becomes an intuitive process that helps make complex algebra easier to manage.