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Polynomial long division is similar to long division we use with numbers but is applied instead to dividing polynomials. This method is especially useful when we want to divide a higher degree polynomial by a first or second degree polynomial. The process involves dividing the leading term of the numerator polynomial by the leading term of the denominator polynomial, multiplying, subtracting, and then bringing down the next term, in a methodical process until the remainder is zero or a polynomial of a lesser degree than the divisor.

For example, if you were to divide \( 4x^3 + 2x^2 - 1 \) by \( x - 3 \), you would divide the \( 4x^3 \) by \( x \) to get \( 4x^2 \) and proceed until you have processed all terms of the numerator or until the remainder is of lower degree than the divisor. In our exercise, we circumvented the need for long division by multiplying with the reciprocal instead.

Factoring polynomials is a foundational technique used to simplify algebraic expressions and solve polynomial equations. The goal is to break down a polynomial into a product of simpler polynomials whose multiplication gives the original polynomial. This can involve factoring by common factors, using difference of squares, factoring trinomials, or more advanced methods like grouping.

In our exercise, we utilize factoring by looking for common factors. We factor out a 2 from the numerator polynomial \(4x^{2}+10\) to get \(2(2x^{2} +5)\) and we recognize that \(x^2 - 9\) is a difference of squares, which can be factored into \( (x - 3)(x + 3) \). Similarly, we factor out a 3 from \(6x^{2}+15\) to simplify to \(3(2x^{2} + 5)\). Understanding and applying these techniques allowed us to manipulate the original division problem into a much simpler form.

Simplifying algebraic expressions is the process of making them easier to understand and work with. This can involve combining like terms, reducing fractions by removing common factors, and more. The primary goal in simplification is to reduce the complexity without changing the value of the expression.

In the exercise, after multiplying by the reciprocal and factoring each polynomial, we simplify the expression by cancelling out common terms. Notice that \(2x^{2} +5\) appears in both the numerator and the denominator and can thus be cancelled out. Furthermore, \(x - 3\) is in the denominator of the first expression and in the numerator of the second, which also allows for cancellation. This process leaves us with a simplified expression \(\frac{2 \times (x + 3)}{3}\). Remember that simplification helps in reducing the risk of errors in further calculations and provides a clearer view of the mathematical relationships within an expression.