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The problem asks to find a second-degree polynomial (quadratic function) that has roots at \( x = -2 \) and \( x = 5 \), and its parabola passes through the point \( (-1, -12) \).

### Step-by-step solution:

1. **General form of a quadratic function with known roots**:
   Since the roots of the polynomial are \( x = -2 \) and \( x = 5 \), the function can be written as:
   \[
   f(x) = a(x + 2)(x - 5)
   \]
   where \( a \) is a constant that we need to determine.

2. **Substitute the point \( (-1, -12) \)**:
   We are given that the parabola passes through the point \( (-1, -12) \), so substitute \( x = -1 \) and \( f(x) = -12 \) into the equation:
   \[
   -12 = a(-1 + 2)(-1 - 5)
   \]
   Simplifying:
   \[
   -12 = a(1)(-6)
   \]
   \[
   -12 = -6a
   \]
   Solving for \( a \):
   \[
   a = \frac{-12}{-6} = 2
   \]

3. **Final quadratic function**:
   Now that we know \( a = 2 \), substitute this back into the equation:
   \[
   f(x) = 2(x + 2)(x - 5)
   \]
   Expanding the expression:
   \[
   f(x) = 2(x^2 - 3x - 10)
   \]
   \[
   f(x) = 2x^2 - 6x - 20
   \]

Thus, the quadratic function is:
\[
f(x) = 2x^2 - 6x - 20
\]

Would you like more details or have any other questions?

Here are five related questions:
1. How do you determine the roots of a quadratic equation?
2. What is the geometric significance of the vertex of a parabola?
3. How do you factor a quadratic expression if its roots are not easily identifiable?
4. Can a quadratic function have complex roots? If so, what would its graph look like?
5. How can we determine the axis of symmetry of a parabola from its quadratic equation?

**Tip:** Always verify your result by plugging back given points or conditions into your equation to ensure correctness.

Quelle est la fonction polynomiale du second degré qui s'annule en (-2) et 5 et dont la parabole passe par le point de coordonnées (-1 ; -12) ?

Learn how to determine a second-degree polynomial that has roots at x = -2 and x = 5, and whose parabola passes through the point (-1, -12). This algebra problem covers key quadratic equation concepts, factorization, and substitution. Suitable for students in Grades 9-11.

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