This image shows an example of adding and subtracting polynomials by grouping like terms. Let's go through the solutions step by step.

Example 5:

(a) $(x^2 + 2x + 3) + (5x^2 + 13)$

• Group like terms: $x^2$ with $5x^2$, and $2x$ with no other $x$-terms, and constant terms.
• Combine them: $(x^2 + 5x^2) + (2x) + (3 + 13)$ $= 6x^2 + 2x + 16$

(b) $(24x^3 - x + 1) - (32x^2 - 7x + 15)$

• First distribute the negative sign to the second polynomial: $24x^3 - x + 1 - 32x^2 + 7x - 15$
• Group like terms: $24x^3 + (-32x^2) + (-x + 7x) + (1 - 15)$ $= 24x^3 - 32x^2 + 6x - 14$

(c) $(x^2 + x + 1) + (2x^2 + 2x - 2) - (x^2 - x - 1)$

• First distribute the negative sign: $(x^2 + x + 1) + (2x^2 + 2x - 2) - x^2 + x + 1$
• Group like terms: $(x^2 + 2x^2 - x^2) + (x + 2x + x) + (1 - 2 + 1)$ $= 2x^2 + 4x + 0$ Simplifying: $= 2x^2 + 4x$

If you need further clarification or details, feel free to ask! Here's a list of related questions you might find helpful:

1. What are the key steps to adding and subtracting polynomials?
2. How do you handle subtraction of polynomials with negative terms?
3. How do you factor polynomials once you've combined like terms?
4. Can polynomials with fractions also be added and subtracted?
5. What is the significance of ordering terms in a polynomial?

Tip: Always make sure to organize terms by their degree (powers of $x$) when working with polynomials to avoid errors!