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Simplifying expressions in algebra means to make equations easier to understand and solve. It involves combining like terms and using arithmetic to reduce the expression to its simplest form. The goal is to take a complex equation and turn it into something simpler without changing its value. This often requires several steps, including applying properties like the distributive property, and carefully simplifying each part. For instance, in the problem \(-7(4p + 1)\), we use the distributive property to simplify the expression. Let's go into each step in more detail.

Multiplication in algebra involves multiplying numbers, variables, or a combination of both. It's just like basic arithmetic multiplication but with variables involved. When we see something like \(-7(4p + 1)\), we need to distribute the \-7\ across everything inside the parentheses. Here's how we do that:
* Multiply \-7\ by \4p\. This gives us \(-7 \times 4p = -28p \).
* Then, multiply \-7\ by \1\. This results in \(-7 \times 1 = -7 \).

Always perform each multiplication step by step, and ensure that you handle positive and negative signs correctly. This process is an application of the distributive property, which tells us how to handle a situation where a single term is multiplied by several others inside parentheses.

Combining like terms means to reduce an expression by merging terms that have the same variable raised to the same power. This is crucial for simplifying expressions. Let's look at our example \(-28p - 7\):
* The term \-28p\ contains the variable \p\.
* The term \-7\ is a constant.
Since these terms don’t have the same variables, we can't combine them. Therefore, \-28p - 7\ is already simplified as much as possible. Remember, only terms that are like (same variables and same exponents) can be combined through addition or subtraction. This step ensures your final expression is neat and as simplified as possible.