1. **State the problem:** Simplify the expression $$\frac{4y - 6}{(y - 6)(y + 6)} - \frac{12 - 3y}{6(y - 6)}.$$\n\n2. **Rewrite and factor where possible:**\n- Factor numerator of first fraction: $$4y - 6 = 2(2y - 3).$$\n- Factor numerator of second fraction: $$12 - 3y = 3(4 - y).$$\n\n3. **Rewrite the expression with factored numerators:**\n$$\frac{2(2y - 3)}{(y - 6)(y + 6)} - \frac{3(4 - y)}{6(y - 6)}.$$\n\n4. **Note that $$4 - y = -(y - 4)$$, so rewrite second numerator:**\n$$3(4 - y) = 3[-(y - 4)] = -3(y - 4).$$\n\n5. **Rewrite the expression:**\n$$\frac{2(2y - 3)}{(y - 6)(y + 6)} - \frac{-3(y - 4)}{6(y - 6)} = \frac{2(2y - 3)}{(y - 6)(y + 6)} + \frac{3(y - 4)}{6(y - 6)}.$$\n\n6. **Find common denominator:**\nThe denominators are $(y - 6)(y + 6)$ and $6(y - 6)$. The least common denominator (LCD) is $$6(y - 6)(y + 6).$$\n\n7. **Rewrite each fraction with the LCD:**\n- First fraction numerator multiplied by 6: $$6 \times 2(2y - 3) = 12(2y - 3).$$\n- Second fraction numerator multiplied by $(y + 6)$: $$3(y - 4)(y + 6).$$\n\nExpression becomes:\n$$\frac{12(2y - 3)}{6(y - 6)(y + 6)} + \frac{3(y - 4)(y + 6)}{6(y - 6)(y + 6)} = \frac{12(2y - 3) + 3(y - 4)(y + 6)}{6(y - 6)(y + 6)}.$$\n\n8. **Expand numerators:**\n- $$12(2y - 3) = 24y - 36.$$\n- Expand $$(y - 4)(y + 6) = y^2 + 6y - 4y - 24 = y^2 + 2y - 24.$$\n- Multiply by 3: $$3(y^2 + 2y - 24) = 3y^2 + 6y - 72.$$\n\n9. **Sum the numerators:**\n$$24y - 36 + 3y^2 + 6y - 72 = 3y^2 + (24y + 6y) + (-36 - 72) = 3y^2 + 30y - 108.$$\n\n10. **Rewrite the fraction:**\n$$\frac{3y^2 + 30y - 108}{6(y - 6)(y + 6)}.$$\n\n11. **Factor numerator:**\nFactor out 3:\n$$3(y^2 + 10y - 36).$$\nFactor quadratic $y^2 + 10y - 36$:\nFind factors of -36 that sum to 10: 12 and -3.\nSo, $$y^2 + 10y - 36 = (y + 12)(y - 3).$$\n\n12. **Rewrite numerator:**\n$$3(y + 12)(y - 3).$$\n\n13. **Rewrite entire fraction:**\n$$\frac{3(y + 12)(y - 3)}{6(y - 6)(y + 6)}.$$\n\n14. **Simplify fraction by canceling common factors:**\n$$\frac{\cancel{3}(y + 12)(y - 3)}{\cancel{6}(y - 6)(y + 6)} = \frac{(y + 12)(y - 3)}{2(y - 6)(y + 6)}.$$\n\n15. **Final simplified expression:**\n$$\boxed{\frac{(y + 12)(y - 3)}{2(y - 6)(y + 6)}}.$$