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The given problem is to calculate \( a + b \) under the condition that the division is exact, where the expression is:

\[
\frac{6x^4 + 21x^3 - 11x^2 + ax + b}{2x^2 + 3x - 5}
\]

Since the division is exact, it implies that the remainder of the division must be zero. To solve this, we need to perform polynomial division (or synthetic division) and find the values of \( a \) and \( b \).

Let me perform the polynomial division step by step to determine \( a + b \).The result of the polynomial division yields:

- Quotient: \( 3x^2 + 6x - 7 \)
- Remainder: \( (a + 51)x + (b - 35) \)

Since the division is exact, the remainder must be zero. Therefore, we can set the coefficients of the remainder equal to zero:

1. \( a + 51 = 0 \)
2. \( b - 35 = 0 \)

Solving these:

- \( a = -51 \)
- \( b = 35 \)

Thus, \( a + b = -51 + 35 = -16 \).

The value of \( a + b \) is \( \boxed{-16} \).

Do you want further details or have any questions?

Here are 5 related questions you might explore:
1. How is polynomial long division performed for non-exact divisions?
2. What are the conditions for a division to be exact in algebra?
3. How can synthetic division be used for higher-degree polynomials?
4. Can we use remainder theorem in such cases?
5. How do coefficients affect the behavior of polynomial division?

**Tip:** When dividing polynomials, always check the degree of the numerator and denominator to anticipate the degree of the quotient and the form of the remainder.

Calcular 'a + b' si la división es exacta: \[ \frac{6x^4 + 21x^3 - 11x^2 + ax + b}{2x^2 + 3x - 5} \]

Explore how to calculate the values of a and b in the polynomial division \( \frac{6x^4 + 21x^3 - 11x^2 + ax + b}{2x^2 + 3x - 5} \) when the division is exact. This step-by-step solution demonstrates polynomial division and the use of the remainder theorem, perfect for students in Grades 10-12.

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