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When you see parentheses in a polynomial expression, particularly in a subtraction problem, it's important to manage them correctly before combining any terms. To remove the parentheses properly, especially when a minus sign precedes them, distribute the negative sign to each term inside. For example, in the expression (-5t + 6) - (t^{2} + 3t - 1), the negative sign in front of the second set of parentheses changes the signs of each term inside. This step transforms it to (-5t + 6) - t^{2} - 3t + 1. By understanding how to remove parentheses correctly, you set yourself up for success in the next steps.

Once the parentheses are removed, the next task is to combine like terms. Like terms are terms that contain the same variable raised to the same power. For example, -5t and -3t are like terms because they both contain the variable t to the first power. To combine them, simply add or subtract the coefficients (the numerical parts) together. Here’s how this works in our example: Combine -5t - 3t = -8t. Then, combine the constants (terms without variables): 6 + 1 = 7. Lastly, remember to keep any unique terms, such as the t^2 term in this expression, as they are. So, our combined terms give us: -t^2 - 8t + 7. This step ensures that all similar terms are properly grouped together, streamlining the expression for further operations.

The distributive property is a key concept when dealing with polynomials, especially in expressions involving parentheses. This property states that a(b + c) = ab + ac. In subtraction problems, it’s crucial to distribute the negative sign properly. In our example (-5t + 6) - (t^2 + 3t - 1), the distributive property helps us understand how to handle each term inside the second set of parentheses. By distributing the negative sign, we change each term inside:• (t^2) becomes -t^2• (3t) becomes -3t• (-1) becomes +1. This gives us the transformed expression -5t + 6 - t^2 - 3t + 1. By correctly applying the distributive property, we make sure that every term is accounted for, setting the stage for easily combining like terms and simplifying the expression.