The equation below gives parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion.
\[
x=4 t+5, y=16 t^{2} ;-\infty< t< \infty
\]

Find a Cartesian equation for the particle's path.
\[
y=\square
\]

Step 1 ：The given equations are parametric equations. To find the Cartesian equation, we need to eliminate the parameter t. We can do this by solving one of the equations for t and then substituting it into the other equation.

Step 2 ：Let's solve the equation for x for t: \(x = 4t + 5\). Solving for t gives \(t = \frac{x - 5}{4}\).

Step 3 ：Next, substitute this expression for t into the equation for y: \(y = 16t^2\). This gives \(y = 16\left(\frac{x - 5}{4}\right)^2\).

Step 4 ：Simplify the equation to get the Cartesian equation for the particle's path. The result is a quadratic equation in terms of x.

Step 5 ：Final Answer: The Cartesian equation for the particle's path is \(\boxed{y = 16\left(\frac{x - 5}{4}\right)^2}\).