Let's solve each part of the question step-by-step.

Given: $P(x) = x^2 + 2x + 3 \quad \text{and} \quad Q(x) = 4x^2 + 5x + 6$

Part (a): $3P(x)$

To find $3P(x)$, multiply each term in $P(x)$ by 3: $3P(x) = 3(x^2 + 2x + 3) = 3x^2 + 6x + 9$

Part (b): $P(x) + Q(x)$

To find $P(x) + Q(x)$, add the corresponding terms in $P(x)$ and $Q(x)$: $P(x) + Q(x) = (x^2 + 2x + 3) + (4x^2 + 5x + 6)$ Combine like terms: $P(x) + Q(x) = (x^2 + 4x^2) + (2x + 5x) + (3 + 6) = 5x^2 + 7x + 9$

Part (c): $P(x) - 2Q(x)$

To find $P(x) - 2Q(x)$, first find $2Q(x)$ by multiplying $Q(x)$ by 2: $2Q(x) = 2(4x^2 + 5x + 6) = 8x^2 + 10x + 12$ Now subtract $2Q(x)$ from $P(x)$: $P(x) - 2Q(x) = (x^2 + 2x + 3) - (8x^2 + 10x + 12)$ Combine like terms: $P(x) - 2Q(x) = (x^2 - 8x^2) + (2x - 10x) + (3 - 12) = -7x^2 - 8x - 9$

Part (d): $P(x)Q(x)$

To find $P(x)Q(x)$, multiply $P(x)$ and $Q(x)$: $P(x)Q(x) = (x^2 + 2x + 3)(4x^2 + 5x + 6)$ Use distribution (or the FOIL method for each pair of terms): $P(x)Q(x) = x^2(4x^2 + 5x + 6) + 2x(4x^2 + 5x + 6) + 3(4x^2 + 5x + 6)$ Expanding each term:

1. $x^2 \cdot 4x^2 = 4x^4$
2. $x^2 \cdot 5x = 5x^3$
3. $x^2 \cdot 6 = 6x^2$
4. $2x \cdot 4x^2 = 8x^3$
5. $2x \cdot 5x = 10x^2$
6. $2x \cdot 6 = 12x$
7. $3 \cdot 4x^2 = 12x^2$
8. $3 \cdot 5x = 15x$
9. $3 \cdot 6 = 18$

Now combine like terms: $P(x)Q(x) = 4x^4 + (5x^3 + 8x^3) + (6x^2 + 10x^2 + 12x^2) + (12x + 15x) + 18$ $= 4x^4 + 13x^3 + 28x^2 + 27x + 18$

Let me know if you would like further explanation on any part.

Follow-up Questions:

1. What is the general approach to multiplying two polynomials?
2. How do you determine the degree of a polynomial resulting from multiplication?
3. Can you explain why combining like terms is essential in polynomial operations?
4. How would the result change if we had $P(x)$ and $Q(x)$ with higher degrees?
5. How do you interpret the terms and coefficients in each polynomial product?

Tip:

When multiplying polynomials, it’s helpful to write each step carefully to avoid missing any terms, especially with higher degrees.