1. **Problem statement:** Given the quadratic equation $$12a^2 - 30a - 1 = 0,$$ find the value of $$4a^2 + \frac{1}{36a^2}.$$\n\n2. **Step 1: Understand the problem and what is asked.**\nWe need to find the value of an expression involving $$a^2$$ and its reciprocal squared term, given a quadratic equation satisfied by $$a$$.\n\n3. **Step 2: Use the quadratic equation to find useful relations.**\nThe equation is $$12a^2 - 30a - 1 = 0.$$\nDivide both sides by 12 to simplify:\n$$a^2 - \frac{30}{12}a - \frac{1}{12} = 0 \implies a^2 - \frac{5}{2}a - \frac{1}{12} = 0.$$\n\n4. **Step 3: Express $$a^2$$ in terms of $$a$$:**\nFrom the equation,\n$$a^2 = \frac{5}{2}a + \frac{1}{12}.$$\n\n5. **Step 4: Find $$4a^2$$:**\nMultiply both sides by 4:\n$$4a^2 = 4 \times \left( \frac{5}{2}a + \frac{1}{12} \right) = 10a + \frac{1}{3}.$$\n\n6. **Step 5: Find $$\frac{1}{36a^2}$$:**\nNote that $$36a^2 = (6a)^2,$$ so we want $$\frac{1}{36a^2}.$$\n\n7. **Step 6: Use the quadratic equation to find $$\frac{1}{a}$$ or $$\frac{1}{a^2}$$:**\nRewrite the original equation as\n$$12a^2 - 30a - 1 = 0.$$\nDivide both sides by $$a$$ (assuming $$a \neq 0$$):\n$$12a - 30 - \frac{1}{a} = 0 \implies \frac{1}{a} = 12a - 30.$$\n\n8. **Step 7: Find $$\frac{1}{a^2}$$:**\nSquare both sides:\n$$\left( \frac{1}{a} \right)^2 = (12a - 30)^2 = 144a^2 - 720a + 900.$$\nSo,\n$$\frac{1}{a^2} = 144a^2 - 720a + 900.$$\n\n9. **Step 8: Find $$\frac{1}{36a^2}$$:**\nDivide both sides by 36:\n$$\frac{1}{36a^2} = \frac{144a^2 - 720a + 900}{36} = 4a^2 - 20a + 25.$$\n\n10. **Step 9: Add $$4a^2$$ and $$\frac{1}{36a^2}$$:**\nRecall from Step 4, $$4a^2 = 10a + \frac{1}{3}.$$\nSo,\n$$4a^2 + \frac{1}{36a^2} = (10a + \frac{1}{3}) + (4a^2 - 20a + 25) = 4a^2 - 10a + 25 + \frac{1}{3}.$$\n\n11. **Step 10: Replace $$4a^2$$ again in the expression:**\nFrom Step 4, $$4a^2 = 10a + \frac{1}{3}$$, so substitute back:\n$$4a^2 - 10a + 25 + \frac{1}{3} = (10a + \frac{1}{3}) - 10a + 25 + \frac{1}{3} = \frac{1}{3} + 25 + \frac{1}{3} = 25 + \frac{2}{3} = \frac{75}{3} + \frac{2}{3} = \frac{77}{3}.$$\n\n**Final answer:**\n$$\boxed{\frac{77}{3}}.$$