After finding both x-intercepts, write $f(x)$ in factored form with an unknown leading coefficient. Then use the vertex y-coordinate.

In $g(x)=f(3-x)+4$, replace every $x$ in the equation for $f$ with $3-x$ before applying the vertical shift.

Algebra is faster here because the axis of symmetry and vertex directly determine $f$. Desmos can verify the transformed equation after you find it.

Enter $y=2(x+1)(x-5)$. Confirm that its x-intercepts are $-1$ and $5$ and that its vertex is $(2,-18)$.

Enter $y=2(3-x+1)(3-x-5)+4$. Compare this graph with the answer choices; it coincides with choice B.

The axis of symmetry is halfway between the x-intercepts. Therefore, $\frac{-1+k}{2}=2$, so $k=5$.

Since the x-intercepts are $-1$ and $5$, write $f(x)=a(x+1)(x-5)$. The vertex occurs at $x=2$, so $f(2)=a(3)(-3)=-9a$. Because the vertex y-coordinate is $-18$, $-9a=-18$, and $a=2$.

Substitute $3-x$ into $f$ and add $4$: $g(x)=2(3-x+1)(3-x-5)+4$. Simplifying the factors gives $g(x)=2(x+2)(x-4)+4$.