Let’s solve the problems step by step:

Part (b): Simplify $(x^2 - x + 5) - (3x^2 + x - 1)$

1. Expand and distribute the negative sign to the second group: $(x^2 - x + 5) - (3x^2 + x - 1) = x^2 - x + 5 - 3x^2 - x + 1$

2. Combine like terms:

   • $x^2 - 3x^2 = -2x^2$
   • $-x - x = -2x$
   • $5 + 1 = 6$

   Final result: $-2x^2 - 2x + 6$

Part (c): Simplify $(x^2 - 3x + 1) - (-x^2 - 1)$

1. Expand and distribute the negative sign to the second group: $(x^2 - 3x + 1) - (-x^2 - 1) = x^2 - 3x + 1 + x^2 + 1$

2. Combine like terms:

   • $x^2 + x^2 = 2x^2$
   • $-3x$ remains as is.
   • $1 + 1 = 2$

   Final result: $2x^2 - 3x + 2$

If you would like more details on these steps, let me know! Here are 5 related questions to expand your understanding:

1. How do you factorize quadratic expressions like those in parts (b) and (c)?
2. What is the process for combining terms in polynomial subtraction?
3. Can you graph the simplified results of these expressions to understand their behavior?
4. What happens if you solve these simplified expressions for specific values of $x$?
5. How does polynomial subtraction differ when variables are raised to higher powers?

Tip: Always distribute the negative sign carefully when subtracting expressions; errors often occur at this step.