Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: $(-1,2),(5,2)$; foci: $(-3,2),(7,2)$

Step 1 ：Given the vertices and foci of the hyperbola, we can find the center, \(h\) and \(k\), by averaging the x-coordinates and y-coordinates of the vertices respectively. For vertices \((-1,2)\) and \((5,2)\), we find that \(h = \frac{-1+5}{2} = 2\) and \(k = \frac{2+2}{2} = 2\).

Step 2 ：The distance between the vertices, \(2a\), is the difference between their x-coordinates, so \(a = \frac{5 - (-1)}{2} = 3\).

Step 3 ：The distance between the foci, \(2c\), is the difference between their x-coordinates, so \(c = \frac{7 - (-3)}{2} = 5\).

Step 4 ：We can find \(b\) using the relationship \(c^2 = a^2 + b^2\). Solving for \(b\), we get \(b = \sqrt{c^2 - a^2} = \sqrt{25 - 9} = 4\).

Step 5 ：Substituting these values into the standard form of the equation of a hyperbola, we get \(\frac{(x - 2)^2}{9} - \frac{(y - 2)^2}{16} = 1\).

Step 6 ：Final Answer: The standard form of the equation of the hyperbola with the given characteristics is \(\boxed{\frac{(x - 2)^2}{9} - \frac{(y - 2)^2}{16} = 1}\).