Plug in the $y$-intercept by using $x=0$ to create an equation you can solve for $a$.

Decide whether $a$ should be positive or negative based on whether the parabola opens upward or downward.

Type $y=a(x+1)(x-5)$ so Desmos creates a slider for $a$.

Plot the point $(0,4)$ by typing $(0,4)$.

Move the slider for $a$ until the parabola passes through $(0,4)$ and opens downward (matching the graph). The displayed value of $a$ determines the equation.

The graph shows $x$-intercepts at $x=-1$ and $x=5$, so the quadratic can be written as

for some constant $a$.

The graph shows the $y$-intercept is $(0,4)$, so $f(0)=4$.

Substitute $x=0$ into $f(x)=a(x+1)(x-5)$:

So $a=-\frac{4}{5}$.

Substituting $a=-\frac{4}{5}$ gives

So the correct choice is $-\frac{4}{5}(x+1)(x-5)$.