1. Key Concepts and Formulas

• Quadratic Polynomial and Roots: A quadratic polynomial with roots $x_1$ and $x_2$ can be written as $f(x) = A(x - x_1)(x - x_2)$, where $A$ is a non-zero constant.
• Factor Theorem: If $r$ is a root of a polynomial $f(x)$, then $(x - r)$ is a factor of $f(x)$.
• Sum of Roots: For a quadratic polynomial $ax^2 + bx + c = 0$, the sum of the roots is given by $-\frac{b}{a}$. In the form $A(x-x_1)(x-x_2)$, the sum of the roots is simply $x_1 + x_2$.

2. Step-by-Step Solution

Step 1: Expressing the Quadratic Polynomial using the Given Root

• What and Why: We are given that $f(x)$ is a quadratic polynomial and one of its roots is $-1$. We want to express $f(x)$ in terms of this root and an unknown root using the Factor Theorem.
• Action: Let the other root of $f(x) = 0$ be $\beta$. Then, we can write $f(x)$ as: $f(x) = A(x - (-1))(x - \beta)$ $f(x) = A(x + 1)(x - \beta) \quad \text{... (Equation 1)}$ Here, $A$ is a non-zero constant.

Step 2: Using the Given Condition to Find the Unknown Root

• What and Why: We are given the condition $f(-2) + f(3) = 0$. We will substitute $x = -2$ and $x = 3$ into the expression for $f(x)$ and use the given condition to solve for the unknown root $\beta$.
• Action:
  • First, calculate $f(-2)$ by substituting $x = -2$ into Equation 1: $f(-2) = A((-2) + 1)((-2) - \beta)$ $f(-2) = A(-1)(-2 - \beta)$ $f(-2) = A(2 + \beta) \quad \text{... (Equation 2)}$
  • Next, calculate $f(3)$ by substituting $x = 3$ into Equation 1: $f(3) = A(3 + 1)(3 - \beta)$ $f(3) = A(4)(3 - \beta)$ $f(3) = A(12 - 4\beta) \quad \text{... (Equation 3)}$
  • Now, substitute Equation 2 and Equation 3 into the given condition $f(-2) + f(3) = 0$: $A(2 + \beta) + A(12 - 4\beta) = 0$
  • Since $A \neq 0$, divide the entire equation by $A$: $(2 + \beta) + (12 - 4\beta) = 0$
  • Combine like terms: $14 - 3\beta = 0$
  • Solve for $\beta$: $3\beta = 14$ $\beta = \frac{14}{3}$
  • Therefore, the second root of $f(x) = 0$ is $\frac{14}{3}$.

Step 3: Calculating the Sum of the Roots

• What and Why: We have identified both roots of $f(x) = 0$ as $x_1 = -1$ and $x_2 = \frac{14}{3}$. We now calculate their sum to answer the question.
• Action: Add the two roots together: $\text{Sum of roots} = (-1) + \left(\frac{14}{3}\right)$ $\text{Sum of roots} = -\frac{3}{3} + \frac{14}{3}$ $\text{Sum of roots} = \frac{14 - 3}{3}$ $\text{Sum of roots} = \frac{11}{3}$

3. Tips and Common Mistakes

• Don't Forget the Leading Coefficient 'A': Always include the constant $A$ in the factored form of a polynomial.
• Correct Factored Form: Remember that if $r$ is a root, the factor is $(x-r)$.
• Algebraic Precision: Be careful with signs and calculations.

4. Summary

We expressed the quadratic polynomial $f(x)$ in terms of its roots, using the given root $-1$ and an unknown root $\beta$. We then used the condition $f(-2) + f(3) = 0$ to solve for $\beta$, obtaining $\beta = \frac{14}{3}$. Finally, we calculated the sum of the roots as $-1 + \frac{14}{3} = \frac{11}{3}$.

The final answer is \boxed{\frac{11}{3}}, which corresponds to option (A).