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Polynomial division is a method used to divide polynomials, much like how long division is applied to numbers. It helps us express a polynomial as a quotient and a remainder. To perform polynomial division, we split the polynomial into smaller, more manageable parts. These parts can be divided using either long division or synthetic division methods. In our exercise, synthetic division is utilized because it simplifies calculations, especially when dividing by a linear polynomial, such as \(x + 3\). The process involves aligning terms by descending powers and carefully subtracting to find each successive term of the quotient. This division yields both the quotient and any remainder, similar to traditional number division.

The Remainder Theorem is a helpful concept in polynomial division that relates to the result of the division. It states that if a polynomial \(f(x)\) is divided by a linear divisor \(x - a\), the remainder of this division is \(f(a)\). This allows us to substitute the value of \(a\) into the polynomial \(f(x)\) to find the remainder quickly. In our case, the divisor is \(x+3\), hence \(a\) is \(-3\) (since we use the opposite sign). The final number in the synthetic division process provides the remainder directly. Here, after completing the synthetic division, the remainder is \(-12\). Hence, it confirms the theorem as substituting \(-3\) in the polynomial should yield the same remainder \(-12\).

Rational expressions can involve one polynomial divided by another. In this context, they're key when discussing the quotient and remainder from polynomial division. Once a polynomial \((5x^3 + 18x^2 + 7x - 6)\) is divided by \((x + 3)\), the outcome is a rational expression, \(5x^2 + 3x - 2 - \frac{12}{x+3}\). This expression showcases the decomposition into a simpler polynomial plus a fractional part based on the remainder and the original divisor. Rational expressions are often simplified to disclose valuable characteristics of functions or equations, and they play a crucial role in calculus and algebraic operations as a bridge to more complex functions.