To find the quadratic equation in intercept form, we start from the x-intercepts of the parabola, which are given as \( (-1, 0) \) and \( (4, 0) \).
The intercept form of a quadratic equation can be written as:
\[ y = a(x + 1)(x - 4) \]
Next, we need to determine the value of \( a \). We can use the third point \( (5, 2) \) to find \( a \). Substituting \( x = 5 \) and \( y = 2 \) into the equation gives us:
\[ 2 = a(5 + 1)(5 - 4) \]
Calculating the terms inside the parentheses:
\[ 2 = a(6)(1) \]
Which simplifies to:
\[ 2 = 6a \]
Solving for \( a \):
\[ a = \frac{2}{6} = \frac{1}{3} \]
Thus, the quadratic equation in intercept form is:
\[ y = \frac{1}{3}(x + 1)(x - 4) \]
None of the provided options have \( \frac{1}{3} \) for \( a \) with intercepts \( (-1,0) \) and \( (4,0) \). However, it could be that the intercepts were misidentified in the calculations.
We can also re-evaluate the options according to the intercept points. The correct option based on calculation outcomes and evaluated positions would be:
\( y = \frac{1}{3}(x + 1)(x - 4) \).
None of the choices given in the response match; it seems there might be an error in the options provided. However, based on the workings outlined above, the equation should be:
\[ y = \frac{1}{3}(x + 1)(x - 4) \]
None of the options directly represent the correct equation per the intercept points and a coordinate check would validate them based on \( a = \frac{1}{3} \).