Yes, that is correct except for a small typo.

At x=1, f'(x)=0.

x=1, y=3. The tangent at x=1 is parallel to the x-axis

(it's a local minimum).

If this tangent is tangent at another point, then there is a 2nd turning point at y=3.

Hence, we find a second solution for x, if y=3 corresponds to f'(x)=0 for more than one x.

This also means we can examine f(x) to see how many x causes f(x)=3.

f(x)=3 for x=1 and x=-2 only.

To check the tangent, if x=-2, f'(x)=-32+16+16+8-4-4=0.

Hence, The tangent is also tangent to the curve at (-2,3).