To find two numbers with a sum of 11 and a product of -312, we can start by letting the two numbers be \( m \) and \( n \). We know:
- \( m + n = 11 \)
- \( m \cdot n = -312 \)
To express this using a quadratic equation, we can relate the sum and product of the roots (the numbers) to the coefficients of the quadratic.
If we assume that \( m \) and \( n \) are the roots of the quadratic equation, we can express this as:
\[ x^2 - (m+n)x + (m \cdot n) = 0 \]
Substituting the known values:
- The sum \( m+n = 11 \), so we have \(-11x\).
- The product \( m \cdot n = -312 \), which gives us \( -312 \).
Thus the quadratic equation becomes:
\[ x^2 - 11x - 312 = 0 \]
Rearranging this gives:
\[ x^2 + 11x - 312 = 0 \]
Now, the quadratic equation can be represented in terms of a standard form. To follow the context of your question, we can set this in the variable \( m \):
\[ m^2 - 11m - 312 = 0 \]
However, we have to check your provided formulas, and it appears that you've presented them in a slightly different format. Based on the context and observing the equation \( m^2 + 11m - 312 \), the term should match because we can think of it as:
\[ m^2 - 11m - 312 \quad (\text{first } \to \text{sum = -11}) \text{ but represented in a way the sum is given as a positive.} \]
Your answer:
- The formula that best represents determining the two numbers is \( m^2 + 11m - 312 = 0 \).
Given that, it appears the option you would select that aligns best with the quadratic form derived from the sum and product requirements would be:
\[ m^2 + 11m - 312 \]
Keep in mind: If you meant to select among your answer options, then the choice correctly aligning should be:
\(\mathbf{m^2 + 11m - 312}\)