Firstly, let's factor the given polynomial: 2x^2 + 11x + 12.
This can be factored by finding two numbers that multiply to give 24 (2 * 12) and add up to 11. These two numbers are 8 and 3.
So, the polynomial factored will be: 2(x + 4)(x + 3 / 2)
Next, let's find the average rate of change from 0 to 3.
Average rate of change is defined as the change in the output values divided by the change in the input values.
Plug these x values (0 and 3) into the function to get the corresponding f(x) or y values.
f(3) = 2(3 + 4)(3 + 1.5) = 44
f(0) = 2(0 + 4)(0 + 1.5) = 12
So, the average rate of change equals (f(3) - f(0)) / (3 - 0) = (44 - 12) / 3 = 10.67
Now, let's find the vertex of the polynomial: 3x^2+18x+9.
Vertex form of a polynomial is y=a(x-h)² +k, where (h, k) is the vertex.
The x-coordinate of the vertex (h) can be found by the formula -b/2a. So, h = -18 / (2* 3) = -3.
Substitute -3 into the given equation to find y-coordinate or k.
k = 3(-3)^2 + 18*-3 + 9 = -9
Hence, the vertex is (-3, -9)
Finally, let's solve the equation: 2x^2 - 104 = 24
First, simplify the equation: 2x^2 - 104 - 24 = 0
that's: 2x^2 - 128 = 0
Divide through by 2 to get: x^2 - 64 = 0
This can be factored to obtain: (x - 8)(x + 8) = 0
Setting each factor equal to zero gives the solutions x = 8 and x = -8.
So, the solution to the equation 2x^2 - 104 = 24 are x = 8 and -8.