Let the four consecutive odd integers be represented by:
\[ x, x + 2, x + 4, x + 6 \]
where \( x \) is the smallest integer. According to the problem, their sum is given as:
\[ x + (x + 2) + (x + 4) + (x + 6) = -288 \]
Now, we can combine like terms:
\[ 4x + 12 = -288 \]
Next, we will isolate \( x \) by first subtracting 12 from both sides:
\[ 4x = -288 - 12 \] \[ 4x = -300 \]
Now, divide both sides by 4 to solve for \( x \):
\[ x = -75 \]
Thus, the four consecutive odd integers are:
\[ -75, -73, -71, -69 \]
To find the value of the least integer, we simply take the value of \( x \):
\[ \text{Least integer} = -75 \]
Lastly, we can verify the sum to ensure correctness:
\[ -75 + (-73) + (-71) + (-69) = -75 - 73 - 71 - 69 \] Calculating step by step:
\[ -75 - 73 = -148 \] \[ -148 - 71 = -219 \] \[ -219 - 69 = -288 \]
The calculation confirms that the total is indeed \(-288\).
Therefore, the value of the least integer is:
\[ \boxed{-75} \]