1. The problem is to solve the equation $$\frac{2x+3}{x-1} = 4.$$\n\n2. The formula used here is to solve rational equations by eliminating the denominator: multiply both sides by the denominator to clear the fraction.\n\n3. Multiply both sides by $x-1$ to get rid of the denominator:\n$$\cancel{\frac{2x+3}{x-1}} \times (x-1) = 4 \times (x-1)$$\nwhich simplifies to\n$$2x + 3 = 4(x - 1).$$\n\n4. Expand the right side:\n$$2x + 3 = 4x - 4.$$\n\n5. Bring all terms to one side to isolate $x$:\n$$2x + 3 - 4x + 4 = 0$$\nwhich simplifies to\n$$-2x + 7 = 0.$$\n\n6. Solve for $x$:\n$$-2x = -7$$\n$$x = \frac{-7}{-2} = \frac{7}{2}.$$\n\n7. Check that $x=\frac{7}{2}$ does not make the denominator zero. Since $x-1 \neq 0$ for $x=\frac{7}{2}$, the solution is valid.\n\nFinal answer: $$x = \frac{7}{2}.$$