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The distributive property helps in simplifying algebraic expressions. This property states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately. For example, for an expression like: \(a(b + c)\), applying the distributive property means: \(a(b + c) = ab + ac\).
Similarly, if you have a negative sign outside the parentheses, like \(-a(b + c)\) it’s the same as \(-ab - ac\).
In our example, consider the first part of the exercise expression: \(-4(-3x+3)\). Using the distributive property: \[-4 \times -3x + (-4) \times 3 = 12x - 12\], and similarly the second part: \[-(6x - 4) = -6x + 4\]. These transformations make the expression easier to handle.

Once you've used the distributive property, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, \(3x\) and \(5x\) are like terms, but \(3x\) and \(3y\) are not because they involve different variables.
In our example, after distributing, our expression looks like this: \[12x - 12 - 6x + 4 - 2x + 1\]. To combine the like terms, first group the x-terms and the constant terms separately:

• \[12x - 6x - 2x \]
• \[-12 + 4 + 1 \]

Combine the x-terms to get: \[12x - 6x - 2x = 4x \], and the constants: \[-12 + 4 + 1 = -7 \], yielding the new, simplified expression: \[4x - 7\].

To make it easy to follow the entire process, here are the algebra steps together. They will help you simplify any complex algebraic expression:

• **Step 1**: Identify and apply the distributive property where necessary.
• **Step 2**: Combine any like terms by grouping similar terms together.
• **Step 3**: Simplify the expression by adding or subtracting the grouped terms.

For our example, let's review the steps:

• **Step 1**: Simplify each part using the distributive property: \[12x - 12 + (-6x + 4) - 2x + 1\]
• **Step 2**: Combine like terms. First the x-terms: \[12x - 6x - 2x = 4x\]. Then the constants: \[-12 + 4 + 1 = -7\].
• **Step 3**: Write the final simplified expression: \[4x - 7\].

Always take your time with each step to ensure accuracy. Practice is key to mastering these concepts.