The multiplicity of a factor in a polynomial indicates how many times that factor appears in the polynomial. For the factor \((x + 1)\) to be part of \(m(x)\):
- A multiplicity of -1 is not possible, as multiplicity must be a non-negative integer.
- A multiplicity of 0 would mean that \((x + 1)\) is not a factor of \(m(x)\) at all, which contradicts the given condition that \((x + 1)\) is a factor.
- A multiplicity of 1 implies that \((x + 1)\) is a factor that appears once in the polynomial, meaning that \(m(-1) = 0\) and the polynomial touches the x-axis at this point.
- A multiplicity of 2 means that \((x + 1)\) is a factor that appears twice in the polynomial, also leading to \(m(-1) = 0\) but indicating that the polynomial touches the x-axis and turns around at this point.
Thus, the possible multiplicities of \((x + 1)\) given that it is a factor of \(m(x)\) are 1 or 2.
Therefore, the valid options for the multiplicity of \((x + 1)\) are:
- Option #3: 1
- Option #4: 2
The other options, -1 and 0, are not valid.