1. **State the problem:** We have a cubic function $p(x) = 3m(x+1)^2$ with a y-intercept of $-12$. We need to find the missing factor $m$. 2. **Recall the y-intercept:** The y-intercept occurs when $x=0$. Substitute $x=0$ into the function: $$p(0) = 3m(0+1)^2 = 3m(1)^2 = 3m$$ 3. **Use the given y-intercept value:** We know $p(0) = -12$, so: $$3m = -12$$ 4. **Solve for $m$:** $$m = \frac{-12}{3}$$ $$m = -4$$ 5. **Interpret the result:** The missing factor $m$ is $-4$. Among the options, $m = x - 4$ is the only one that matches the value $-4$ when $x=0$. 6. **Check the options:** Since $m$ is a factor, it should be a constant or a function that when multiplied by $3(x+1)^2$ gives a cubic polynomial. The only option that fits the numeric value $-4$ is $m = x - 4$. **Final answer:** $m = x - 4$ (option a)