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Subtracting polynomials might sound daunting, but it's actually similar to the subtraction you do with normal numbers - with one key difference: you deal with variables raised to powers.

When subtracting one polynomial from another, like \( (-2 x^{3}+x-1) - (-x^{2}+x-3) \), it's crucial to understand that you operate on each term separately. A term in a polynomial is a number, a variable, or numbers and variables multiplied together. Essentially, you're looking to find the difference between like terms – those that have the same variable raised to the same power.

To get there, you first distribute the subtraction sign across the polynomial you're subtracting, changing each term's sign. This is actually turning the subtraction problem into an addition problem - with signs adjusted. You would transform the example into \( (-2x^{3} + x - 1) + (x^{2} - x + 3) \).

After that, it's a matter of simplifying the expression, which leads us to our next topic - combining like terms.

Once subtraction is converted to addition and terms are laid out clearly, the next step is combining like terms. In algebra, like terms are terms that have the same variables and powers. The coefficients (the numbers in front of the variables) can be different.

Let's revisit our example. After we've adjusted the signs, we get \[ -2x^{3} + x - 1 + x^{2} - x + 3 \]. Notice how \( x \) and \( -x \) are like terms. They cancel each other out because adding a number to its opposite equals zero.

This leaves us with \( -2x^{3} + x^{2} + 2 \). \( -2x^{3} \) and \( x^{2} \) aren't like terms because they have different powers, so they stay as they are. However, the constants \( -1 \) and \( 3 \) are like terms (since they have no variable at all!) and when combined give us \( 2 \) as their sum. Through this process, we systematically removed parentheses and combined what could be combined.

Using the horizontal format in algebra refers to writing and solving problems in a linear, side-by-side manner. It's the way we've set up our subtraction problem: \( (-2x^{3} + x - 1) - (-x^{2} + x - 3) \).

This view makes it easier for you to see all the terms at once and helps avoid mistakes when combining like terms. You can directly compare coefficients and keep track of variable terms and their corresponding exponents, essentially streamlining the whole process.

While vertical format stacks the terms above each other, which can be helpful for addition and multiplication of polynomials, horizontal format is typically preferred for subtraction to clearly show each subtraction change, making it straightforward to distribute the negative sign across the second polynomial. This approach ensures a clean and less error-prone path to the solution for polynomial subtraction problems.