Solutions:

---

1. Subtract $2x^2 + x + 1$ from $4x^2 + 3x + 5$:
$(4x^2 + 3x + 5) - (2x^2 + x + 1) = 4x^2 - 2x^2 + 3x - x + 5 - 1 = 2x^2 + 2x + 4.$
Answer: $\mathbf{a. \ 2x^2 + 2x + 4}$.

---

2. Add $3x^2 + 4x + 5$ and $2x^2 + x + 3$:
$(3x^2 + 4x + 5) + (2x^2 + x + 3) = 3x^2 + 2x^2 + 4x + x + 5 + 3 = 5x^2 + 5x + 8.$
Answer: $\mathbf{a. \ 5x^2 + 5x + 8}$.

---

3. Add $4x^2 + 2x + 3$ and $x^2 + 3x + 5$:
$(4x^2 + 2x + 3) + (x^2 + 3x + 5) = 4x^2 + x^2 + 2x + 3x + 3 + 5 = 5x^2 + 5x + 8.$
Answer: $\mathbf{a. \ 5x^2 + 5x + 8}$.

---

4. Subtract $2x^2 + 3x + 4$ from $5x^2 + 6x + 7$:
$(5x^2 + 6x + 7) - (2x^2 + 3x + 4) = 5x^2 - 2x^2 + 6x - 3x + 7 - 4 = 3x^2 + 3x + 3.$
Answer: $\mathbf{d. \ 3x^2 + 3x + 3}$.

---

5. Add $x^2 + 4x + 4$ and $2x^2 + 3x + 1$:
$(x^2 + 4x + 4) + (2x^2 + 3x + 1) = x^2 + 2x^2 + 4x + 3x + 4 + 1 = 3x^2 + 7x + 5.$
Answer: $\mathbf{c. \ 3x^2 + 7x + 5}$.

---

6. Add $3x^2 + 2x + 1$ and $2x^2 + 4x + 3$:
$(3x^2 + 2x + 1) + (2x^2 + 4x + 3) = 3x^2 + 2x^2 + 2x + 4x + 1 + 3 = 5x^2 + 6x + 4.$
Answer: $\mathbf{b. \ 5x^2 + 6x + 4}$.

---

7. Product of $(2x + 5)$ and $(x - 3)$:
Using distributive property: $(2x + 5)(x - 3) = 2x \cdot x + 2x \cdot (-3) + 5 \cdot x + 5 \cdot (-3) = 2x^2 - 6x + 5x - 15 = 2x^2 - x - 15.$
Answer: $\mathbf{a. \ 2x^2 - x - 15}$.

---

8. Add $2x^2 + 6x + 4$ and $3x^2 + 4x + 5$:
$(2x^2 + 6x + 4) + (3x^2 + 4x + 5) = 2x^2 + 3x^2 + 6x + 4x + 4 + 5 = 5x^2 + 10x + 9.$
Answer: $\mathbf{c. \ 5x^2 + 10x + 9}$.

---

9. Product of $(x - 4)$ and $(x + 3)$:
Using distributive property: $(x - 4)(x + 3) = x \cdot x + x \cdot 3 + (-4) \cdot x + (-4) \cdot 3 = x^2 + 3x - 4x - 12 = x^2 - x - 12.$
Answer: $\mathbf{a. \ x^2 - x - 12}$.

---

10. Subtract $2x^2 + 4x + 6$ from $6x^2 + 5x + 7$:
$(6x^2 + 5x + 7) - (2x^2 + 4x + 6) = 6x^2 - 2x^2 + 5x - 4x + 7 - 6 = 4x^2 + x + 1.$
Answer: $\mathbf{c. \ 4x^2 + x + 1}$.

---

Follow-up Questions:

1. Can you explain why combining like terms is essential in polynomial operations?
2. How does the distributive property simplify the multiplication of binomials?
3. What strategies can help avoid errors when subtracting polynomials?
4. How can you check your work after performing polynomial addition or subtraction?
5. What are the common mistakes students make when working with polynomials, and how can they be avoided?

Tip:

Always write each step clearly when working with polynomials to avoid missing terms or signs!