1. The problem is to solve the equation $$\frac{2x+3}{x-1} = 4.$$\n\n2. We use the property that if $$\frac{a}{b} = c$$ and $$b \neq 0$$, then $$a = bc$$. Here, $$a = 2x+3$$, $$b = x-1$$, and $$c = 4$$.\n\n3. Multiply both sides by $$x-1$$ to eliminate 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. Rearrange terms to isolate $$x$$:\n$$2x + 3 - 4x = -4$$\n$$\cancel{2x} + 3 - \cancel{4x} = -4$$\n$$-2x + 3 = -4.$$\n\n6. Subtract 3 from both sides:\n$$-2x + 3 - 3 = -4 - 3$$\n$$-2x = -7.$$\n\n7. Divide both sides by $$-2$$ to solve for $$x$$:\n$$x = \frac{-7}{-2} = \frac{7}{2}.$$\n\n8. The solution is $$x = \frac{7}{2} = 3.5.$$