We are given the following polynomial equations:

1. $p(x) = 4x^5 + ax^2 + (a - 3)x + 3$
2. $q(x) = 2x^4 - x^3 + 2bx^2 + (2b + 1)x + 1$
3. $r(x) = 4x^5 + 2x^4 - x^3 + 5x^2 + 3x + 4$

We need to determine the values of $a$ and $b$ by solving for $p(x) + q(x) = r(x)$.

Step-by-Step Breakdown:

Start by adding $p(x)$ and $q(x)$ together:

$p(x) + q(x) = (4x^5 + ax^2 + (a - 3)x + 3) + (2x^4 - x^3 + 2bx^2 + (2b + 1)x + 1)$

Combine like terms:

$p(x) + q(x) = 4x^5 + 2x^4 - x^3 + (a + 2b)x^2 + [(a - 3) + (2b + 1)]x + (3 + 1)$

This simplifies to:

$p(x) + q(x) = 4x^5 + 2x^4 - x^3 + (a + 2b)x^2 + (a + 2b - 2)x + 4$

Now, compare the coefficients of this polynomial with $r(x) = 4x^5 + 2x^4 - x^3 + 5x^2 + 3x + 4$.

From the comparison:

1. For the $x^2$-term: $a + 2b = 5$
2. For the $x$-term: $a + 2b - 2 = 3$, which simplifies to $a + 2b = 5$ (this matches the equation above).
3. For the constant term: $4 = 4$ (this is already satisfied).

Thus, we need to solve $a + 2b = 5$.

Since there is only one equation, we can assume $a$ and $b$ can take various pairs of values that satisfy this equation. However, it's common to solve these types of problems by assuming one of the variables and solving for the other.

Would you like to proceed with an assumption for either $a$ or $b$ to determine their values, or do you want to explore all possible solutions?

5 Further Questions:

1. What assumptions might simplify the system in determining values for $a$ and $b$?
2. Are there any restrictions on the values of $a$ and $b$ in this context?
3. How would changing the degree of one of the polynomials affect the process of matching coefficients?
4. Can you solve for more than two variables using this method of coefficient comparison?
5. What are the implications if a polynomial equation is not fully defined by the given coefficients?

Tip:

Always remember to check all terms in polynomial addition, especially for higher degrees (like $x^5$ or $x^4$), as these terms guide the comparison of the two sides effectively.