Given information:
- The quadratic equation is given by F(x) = 0.
- F(-π) = F(π) = 0.
- F(π/2) = -3π^2/4.

To find:
- The value of lim x → -π F(x)/sin(sinx).

Solution:

Step 1: Finding the quadratic equation:
Since F(-π) = F(π) = 0, we know that -π and π are the roots of the quadratic equation. Therefore, the quadratic equation can be written in the form:
F(x) = a(x + π)(x - π)

Step 2: Finding the value of a:
To find the value of 'a', we can substitute the given point F(π/2) = -3π^2/4 into the equation:
F(π/2) = a(π/2 + π)(π/2 - π) = -3π^2/4
Simplifying the equation, we get:
a(3π/2)(-π/2) = -3π^2/4
-3a(π^2/4) = -3π^2/4
-3a = -1
a = 1/3

Therefore, the quadratic equation becomes:
F(x) = (1/3)(x + π)(x - π)

Step 3: Finding the limit:
We need to find the value of lim x → -π F(x)/sin(sinx).

Let's simplify the expression:
F(x)/sin(sinx) = [(1/3)(x + π)(x - π)] / sin(sinx)

Since sin(sinx) is a continuous function, we can rewrite the limit as:
lim x → -π [(1/3)(x + π)(x - π)] / sin(sinx)
= [(1/3)(-π + π)(-π - π)] / sin(sin(-π))
= [(1/3)(-2π)(-2π)] / sin(0)
= (4π^2/3) / 0

Step 4: Finding the value of the limit:
To find the value of the limit, we need to examine the behavior of the function as x approaches -π from the left and right sides.

4.1. Left-hand limit:
As x approaches -π from the left side, the function F(x)/sin(sinx) approaches negative infinity. This is because sin(sinx) approaches 0 from the left side, and the numerator approaches a negative value (-4π^2/3).

4.2. Right-hand limit:
As x approaches -π from the right side, the function F(x)/sin(sinx) approaches positive infinity. This is because sin(sinx) approaches 0 from the right side, and the numerator approaches a positive value (-4π^2/3).

Step 5: Conclusion:
Since the left-hand limit and the right-hand limit of the function F(x)/sin(sinx) approach different values (negative infinity and positive infinity), the limit does not exist. Therefore, we can conclude that the value of lim x → -π