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When simplifying algebraic expressions, one of the fundamental tools to understand is the distributive property. The distributive property allows you to multiply a single term by each term in a parenthesis. This means if you have an expression like \(a(b + c)\), you can distribute \(a\) to both \(b\) and \(c\), resulting in \(ab + ac\). For example, in the expression given in the exercise, we have to distribute \(-6\) to each term inside the parentheses: \(-6(2a^2 - 4ab - 10b^2)\). Applying the distributive property, you get \(-6 \times 2a^2 = -12a^2\), \(-6 \times (-4ab) = 24ab\), and \(-6 \times (-10b^2) = 60b^2\). Putting it all together: \(-12a^2 + 24ab + 60b^2\). Following this rule makes the next steps straightforward.

In algebra, combining like terms is an essential step to simplify expressions. Like terms are terms that have the same variable raised to the same power. For example, \(-8a^2\) and \(-12a^2\) are like terms, but \(a\) and \(a^2\) are not. To combine like terms, you simply add or subtract their coefficients (the numerical parts). In the exercise, after distributing the \(-6\), we had \(-8a^2 + 5ab - 12b^2 - 12a^2 + 24ab + 60b^2\). Grouping the like terms, we get \((-8a^2 - 12a^2) + (5ab + 24ab) + (-12b^2 + 60b^2)\). Now combine them: \(-8a^2 - 12a^2 = -20a^2\), \(5ab + 24ab = 29ab\), and \(-12b^2 + 60b^2 = 48b^2\). Simplifying like this makes the entire expression much more manageable.

Polynomial expressions consist of variables and coefficients, combined using addition, subtraction, and multiplication. They are built from terms like \(a^2\), \(ab\), or \(b^2\). A key part of working with polynomials is simplifying them to their least complex form by using algebraic properties and rules. In the exercise, we started with a polynomial expression: \(-8a^2 + 5ab - 12b^2 - 6(2a^2 - 4ab - 10b^2)\). The goal was to simplify this expression through distributing and combining like terms. After careful simplification, we arrived at the simplified polynomial: \(-20a^2 + 29ab + 48b^2\). Understanding how to manipulate and simplify polynomials is crucial for solving more complex algebraic problems.