To solve the problem of generating all possible ways to add parentheses to a mathematical expression (e.g., “2-1-1” or “2*3-4*5″), we need to consider how to evaluate expressions in multiple orders using recursive partitioning. Here’s a solution in JavaScript:

### Problem:

Given a string containing numbers and operators (`+, -, *`

), return all possible results from computing all the different possible ways to group numbers and operators.

### Approach:

**Recursive Partitioning**: For each operator, partition the expression into left and right sides.**Recursive Evaluation**: Recursively compute the result of the left and right subexpressions.**Combine**: Combine the results from the left and right sides using the operator.

### Solution Code:

```
function diffWaysToCompute(expression) {
const results = [];
// Base case: if the expression is a single number, return it as the only result
if (!expression.match(/[\+\-\*]/)) {
return [parseInt(expression)];
}
// Loop through each character in the expression
for (let i = 0; i < expression.length; i++) {
const char = expression[i];
// If the character is an operator
if (char === '+' || char === '-' || char === '*') {
// Recursively compute the left and right side of the expression
const leftResults = diffWaysToCompute(expression.slice(0, i));
const rightResults = diffWaysToCompute(expression.slice(i + 1));
// Combine the results from the left and right sides using the current operator
for (const left of leftResults) {
for (const right of rightResults) {
let result;
if (char === '+') {
result = left + right;
} else if (char === '-') {
result = left - right;
} else if (char === '*') {
result = left * right;
}
results.push(result);
}
}
}
}
return results;
}
// Example usage:
const expression = "2*3-4*5";
console.log(diffWaysToCompute(expression)); // Output: [-34, -10, -14, -10, 10]
```

### Explanation:

**Base Case**: If the`expression`

contains no operators, it’s a number, and we return it.**Recursive Step**: For each operator (`+, -, *`

), split the expression into two parts: left and right.**Recursive Call**: Recursively compute possible results for both parts.**Combine Results**: After the recursive calls, we combine the results from the left and right parts using the operator.

### Example:

For the input `"2*3-4*5"`

, the possible ways to compute it are:

`(2 * (3 - (4 * 5))) = -34`

`((2 * 3) - (4 * 5)) = -14`

- And more combinations.

This solution uses recursion and divides the problem into subproblems, making it an efficient way to explore all valid groupings of parentheses.