Show that the equation `x^(3)+2x^(2)+x+5=0` has only one real root, such that `[alpha]=-3`, where `[x]` denotes the integral point of `x`

Explanation

To show that the equation \( x^3 + 2x^2 + x + 5 = 0 \) has only one real root such that the greatest integer function \( [\alpha] = -3 \), we can follow these steps:

### Step 1: Define the function
Let \( f(x) = x^3 + 2x^2 + x + 5 \).

### Step 2: Evaluate the function at the endpoints of the interval
We need to evaluate \( f(x) \) at \( x = -3 \) and \( x = -2 \):

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