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Factoring polynomials is a fundamental skill in algebra and is especially important when working with rational expressions. In its simplest terms, factoring involves breaking down a polynomial into a product of its simpler components. These components are often called factors, and they multiply together to give the original polynomial.

When you're asked to factor a polynomial, you should first look for the greatest common factor (GCF) of all the terms. The GCF is the largest expression that divides each term in the polynomial completely. In our example, with the numerator \(9x^2 + 9x\), the GCF is \(9x\) because every term shares this factor.

Once you factor out the GCF, the expression can be rewritten as a multiplication, making it easier to simplify further down the line. Proper factoring is crucial since it paves the way for canceling terms and simplifying rational expressions.

Rational expressions are like fractions but with polynomials in the numerator and denominator. They follow many of the same mathematical rules as numerical fractions. Just like numerical fractions, if you want to simplify a rational expression, you simplify by removing common factors.

For instance, the expression \(\frac{9x^2 + 9x}{2x + 2}\) is a rational expression because both the numerator and denominator are polynomials. The simplification process involves factoring both parts thoroughly and then reducing by canceling out common factors.

While simplifying rational expressions, it's essential to factor both the top (numerator) and the bottom (denominator) fully, so nothing is left to cancel out, ensuring you reach its simplest form. As with fractions, when the simplest form is reached, it cannot be reduced further.

Algebraic fractions are essentially fractions where the numerator and/or the denominator are algebraic expressions. This means that they can include variables, and these variables can be squared, cubed, or even more complex.

The goal when working with algebraic fractions is often to simplify them as much as possible. Simplification involves using factoring and the properties of fractions, such as canceling out common terms both top and bottom. For example, in the exercise, after factoring \(9x(x + 1)\) and \(2(x + 1)\), the common factor \((x + 1)\) is canceled, leaving a simpler algebraic fraction \(\frac{9x}{2}\).

Understanding how algebraic fractions work and their simplification is essential in algebra, as it leads to clearer, more manageable expressions and helps solve equations more easily.