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Distributing a negative sign (or subtracting a polynomial) means you must change the sign of every term inside the parentheses you're subtracting. For the given exercise: \ \( (x^2 + x) - (3x^2 + 2x - 1) \)
First, identify the polynomial you are subtracting. Here, it's \( (3x^2 + 2x - 1) \).
Next, distribute the negative sign to each term inside the parenthesis: \ \( - (3x^2) = -3x^2 \)
\( - (2x) = -2x \)
\( - (-1) = +1 \)
Now, rewrite the expression: \ \( (x^2 + x) - (3x^2 + 2x - 1) = x^2 + x - 3x^2 - 2x + 1 \).
Remember to change the sign of every term inside the parentheses during this operation. This is a crucial step in polynomial operations.

Combining like terms means summing or subtracting terms with the same variable and exponent. Once the negative sign is distributed, you will have: \ \( x^2 + x - 3x^2 - 2x + 1 \).
Identify and combine the like terms: \ \( x^2 - 3x^2 \), both terms have the same variable \( x \) and exponent \( 2 \): \ \( x^2 - 3x^2 = -2x^2 \)
Next, look at the terms with variable \( x \) but exponent \( 1 \): \ \( x - 2x = -x \).
Finally, the constant term is \( +1 \), and it remains as it is.
Hence, after combining like terms, the polynomial simplifies to: \ \( -2x^2 - x + 1 \).
This step helps to simplify the polynomial further by grouping and reducing similar types.

Subtraction of polynomials involves two main steps: distributing any negative signs and combining like terms. After distributing the negative sign and changing the terms, you must combine terms that share the same variable and exponent.
The initial polynomial is: \ \( (x^2 + x) - (3x^2 + 2x - 1) \).
After distributing the negative sign: \ \( x^2 + x - 3x^2 - 2x + 1 \).
Combining like terms: \ \( x^2 - 3x^2 = -2x^2 \)
\( x - 2x = -x \)
So, the final simplified form is: \ \( -2x^2 - x + 1 \).
This shows the critical steps in polynomial subtraction: properly distributing negative signs and efficiently combining like terms. Understanding these processes simplifies handling more complex polynomials.