1. **State the problem:** Simplify and add the expressions $$\frac{16x^2 - 8x + 1}{9x} + \frac{12x + 3}{1 - 16x^2}$$. 2. **Identify the formula and rules:** To add fractions, find a common denominator. Here, denominators are $$9x$$ and $$1 - 16x^2$$. Note that $$1 - 16x^2$$ is a difference of squares: $$1 - (4x)^2 = (1 - 4x)(1 + 4x)$$. 3. **Rewrite the problem with factored denominators:** $$\frac{16x^2 - 8x + 1}{9x} + \frac{12x + 3}{(1 - 4x)(1 + 4x)}$$ 4. **Find the least common denominator (LCD):** LCD = $$9x(1 - 4x)(1 + 4x)$$ 5. **Rewrite each fraction with the LCD:** - Multiply numerator and denominator of the first fraction by $$(1 - 4x)(1 + 4x)$$ - Multiply numerator and denominator of the second fraction by $$9x$$ 6. **Calculate new numerators:** - First numerator: $$(16x^2 - 8x + 1)(1 - 4x)(1 + 4x)$$ - Second numerator: $$(12x + 3)(9x) = 108x^2 + 27x$$ 7. **Simplify $$(1 - 4x)(1 + 4x)$$:** $$(1 - 4x)(1 + 4x) = 1 - 16x^2$$ 8. **Expand the first numerator:** $$(16x^2 - 8x + 1)(1 - 16x^2) = (16x^2 - 8x + 1) - 16x^2(16x^2 - 8x + 1)$$ $$= 16x^2 - 8x + 1 - 256x^4 + 128x^3 - 16x^2$$ $$= -256x^4 + 128x^3 + (16x^2 - 16x^2) - 8x + 1$$ $$= -256x^4 + 128x^3 - 8x + 1$$ 9. **Add the numerators:** $$(-256x^4 + 128x^3 - 8x + 1) + (108x^2 + 27x) = -256x^4 + 128x^3 + 108x^2 + 19x + 1$$ 10. **Write the final expression:** $$\frac{-256x^4 + 128x^3 + 108x^2 + 19x + 1}{9x(1 - 4x)(1 + 4x)}$$ This is the simplified sum of the two rational expressions. **Final answer:** $$\boxed{\frac{-256x^4 + 128x^3 + 108x^2 + 19x + 1}{9x(1 - 4x)(1 + 4x)}}$$