Let’s solve each of the given problems step-by-step:

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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: a. $2x^2 + 2x + 4$

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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: a. $5x^2 + 5x + 8$

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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: a. $5x^2 + 5x + 8$

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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: d. $3x^2 + 3x + 3$

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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: c. $3x^2 + 7x + 5$

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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: b. $5x^2 + 6x + 4$

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7. Find the product of $(2x + 5)$ and $(x - 3)$:

$(2x + 5)(x - 3) = 2x \cdot x + 2x \cdot (-3) + 5 \cdot x + 5 \cdot (-3)$ $= 2x^2 - 6x + 5x - 15 = 2x^2 - x - 15$

Answer: a. $2x^2 - x - 15$

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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: c. $5x^2 + 10x + 9$

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9. Find the product of $(x - 4)$ and $(x + 3)$:

$(x - 4)(x + 3) = x \cdot x + x \cdot 3 - 4 \cdot x - 4 \cdot 3$ $= x^2 + 3x - 4x - 12 = x^2 - x - 12$

Answer: c. $x^2 - x - 12$

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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: c. $4x^2 + x + 1$

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Summary of Answers:

1. a
2. a
3. a
4. d
5. c
6. b
7. a
8. c
9. c
10. c

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Do you have any specific questions on the steps, or would you like details about a particular problem?

Related Questions:

1. How do you subtract polynomials effectively?
2. What are some shortcuts for adding polynomials?
3. What is the FOIL method used in polynomial multiplication?
4. How do you identify and combine like terms in polynomial expressions?
5. What happens when subtracting polynomials with negative coefficients?

Tip: When adding or subtracting polynomials, always align terms with the same degree to avoid mistakes.