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Factoring a polynomial is a method used to break down the expression into simpler components called factors. These factors, when multiplied together, will give back the original polynomial. In the example provided, we start with the polynomial expression \(8c^2 - 8c + 11c - 11\). The main goal of factoring is to identify and rewrite the expression using these factors. This process often makes solving equations easier or simplifies the problems significantly. Key steps in polynomial factoring involve rearranging terms, finding common factors, and using various techniques to break down the expression. For this particular case, the technique of grouping is ideal to simplify the polynomial into a product of binomials.

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. In our exercise, the expression is \(8c^2 - 8c + 11c - 11\). This expression needs to be grouped and simplified.An important property of algebraic expressions is that they can often be manipulated in different ways while maintaining equality. By organizing terms with common factors, expressions can be rearranged to make factoring easier. In the expression given, we have a mix of terms involving the same variable with different exponents. Recognizing which terms to group and factor is a key skill when dealing with algebraic expressions. Simplifying them appropriately paves the way for further mathematical operations, such as solving equations or analyzing functions.

There are multiple techniques for factoring algebraic expressions, and one of the most useful is 'factor by grouping.' This method is particularly effective with polynomials that have four or more terms.Let's break down the process:

• Grouping Terms: First, group terms that share common factors. In our example, we group as \((8c^2 - 8c) + (11c - 11)\).
• Finding Common Factors: Identify the greatest common factor within each group. For \(8c^2 - 8c\), it's \(8c\), and for \(11c - 11\), it's \(11\).
• Factoring Out: Factor out these common factors: \(8c(c - 1) + 11(c - 1)\).
• Common Binomial Factor: Look for a common binomial factor in the newly grouped terms. Here, \((c - 1)\) is common.

Finally, the expression is factored as \((8c + 11)(c - 1)\). This method efficiently reduces complex expressions into more manageable components.