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Combining like terms is a fundamental part of simplifying algebraic expressions. It involves adding or subtracting terms that have the same variable raised to the same power.
For instance, in the problem \[ 4a + 4b + 4a - b \], we have similar terms which are \(4a\) and \(4a\), and then \(4b\) and \(-b\).
To combine these terms, we perform the following:
- Add or subtract the coefficients of like terms.
- \(4a + 4a\) results in \(8a\).
- \(4b - b\) results in \(3b\).
Ultimately, combining like terms simplifies our expression to \(8a + 3b\). Always ensure that you identify terms with the same variables and combine their coefficients accurately.

Distributing coefficients is another crucial concept in simplifying expressions. It means multiplying a coefficient by each term inside parentheses.
In the given expression, \[ 4(a+b) + 4a - b \], we distribute the coefficient \(4\) to both \(a\) and \(b\).
This results in \(4a + 4b\).
Here's how it's done:
- Multiply \(4\) by \(a\) to get \(4a\).
- Multiply \(4\) by \(b\) to get \(4b\).
After distribution, we incorporate the other terms to get: \(4a + 4b + 4a - b\). This step makes it easier to then combine like terms.

When working with algebraic expressions, understanding the basics is key. This includes the operations of addition, subtraction, multiplication, and distribution.
Let's break it down:
- **Terms**: These are parts of an algebraic expression separated by + or - signs.
- **Coefficients**: Numbers multiplying a variable. For example, in \(4a\), 4 is the coefficient.
- **Variables**: These are symbols representing unknown values, commonly \(a, b, x, y\), etc.
- **Constants**: Fixed numbers with no variables attached.
In our exercise, \[4(a+b)+4a-b\], \ 4, and -1 are coefficients, and \(a\) and \(b\) are variables.
Mastery of these basics allows for the correct application of distribution and combining like terms, resulting in simplified expressions like \(8a + 3b\).
Always take it step-by-step, ensuring accuracy in each operation. With practice, these basics become second nature.