texture mapping a trapezoid with a square texture in OpenGL

I've been trying to render a GL_QUAD (which is shaped as a trapezoid) with a square texture. I'd like to try and use OpenGL only to pull this off. Right now the texture is getting heavily distorted and it's really annoying.

Normally, I would load the texture compute a homography but that means a lot of work and an additional linear programming library/direct linear transform function. I'm under the impression OpenGL can simplify this process for me.

I've looked around the web and have seen "Perspective-Correct Texturing, Q Coordinates, and GLSL" and "Skewed/Sheared Texture Mapping in OpenGL".

These all seem to assume you'll do some type of homography computation or use some parts of OpenGL I'm ignorant of ... any advice?

Update:

I've been reading "Navigating Static Environments Using Image-Space Simplification and Morphing" [PDF] - page 9 appendix A.

It looks like they disable perspective correction by multiplying the (s,t,r,q) texture coordinate with the vertex of a model's world space z component.

so for a given texture coordinate (s, r, t, q) for a quad that's shaped as a trapezoid, where the 4 components are:

(0.0f, 0.0f, 0.0f, 1.0f),
(0.0f, 1.0f, 0.0f, 1.0f),
(1.0f, 1.0f, 0.0f, 1.0f),
(1.0f, 0.0f, 0.0f, 1.0f) 

This is as easy as glTexCoord4f (svert.z, rvert.z, t, q*vert.z)? Or am I missing some step? like messing with the GL_TEXTURE glMatrixMode?

Update #2:

That did the trick! Keep it in mind folks, this problem is all over the web and there weren't any easy answers. Most involved directly recalculating the texture with a homography between the original shape and the transformed shape...aka lots of linear algebra and an external BLAS lib dependency.