1. Statement of the problem: We are given the equation $y^2 - 4x + h x = 0$ and we will analyze its conic type and key parameters.\n 2. Formula and important rules: The standard form of a horizontal parabola is $y^2 = 4 p x$ where $p$ is the focal parameter.\n Rules: The parabola opens to the right if $p>0$ and to the left if $p<0$.\n Rules: The vertex is at the origin when there are no translation terms in $x$ or $y$.\n Rules: The focus is at $\left(p,0\right)$ and the directrix is $x=-p$.\n Rules: The length of the latus rectum is $|4p|$.\n 3. Intermediate work — rewrite the equation: Starting from $y^2 - 4x + h x = 0$ we collect $x$-terms to get $y^2 = x(4 - h)$.\n 4. Match to standard form: Compare $y^2 = x(4 - h)$ with $y^2 = 4 p x$ to identify $4 p = 4 - h$.\n 5. Solve for the focal parameter: Therefore $p = \frac{4 - h}{4}$.\n 6. Vertex: Since there are no translation terms the vertex is at the origin $\left(0,0\right)$.\n 7. Focus: The focus is at $\left(p,0\right)$ which equals $\left(\frac{4 - h}{4},0\right)$.\n 8. Directrix: The directrix is $x = -p$ which equals $x = -\frac{4 - h}{4}$.\n 9. Latus rectum: The length of the latus rectum is $|4p| = |4 - h|$ and it is a vertical segment through the focus.\n 10. Orientation and special cases: If $4 - h > 0$ the parabola opens to the right, if $4 - h < 0$ it opens to the left, and if $4 - h = 0$ the equation reduces to $y^2 = 0$ which is the degenerate double line $y=0$.\n 11. Final answer summary: The given equation represents a horizontal parabola with standard form $y^2 = (4 - h)x$, focal parameter $p = \frac{4 - h}{4}$, vertex $\left(0,0\right)$, focus $\left(\frac{4 - h}{4},0\right)$, directrix $x = -\frac{4 - h}{4}$, latus rectum length $|4 - h|$, and orientation determined by the sign of $4 - h$.\n