1. **State the problem:** We are given the equation $$(2x + 3y)^2 - 8x - 12y + 16 = 0$$ and need to find the value of $$A$$ where $$A = 2x + 3y$$. 2. **Rewrite the problem in terms of $$A$$:** Since $$A = 2x + 3y$$, substitute into the equation: $$A^2 - 8x - 12y + 16 = 0$$ 3. **Express $$x$$ and $$y$$ in terms of $$A$$:** We have one equation but two variables, so let's try to express $$-8x - 12y$$ in terms of $$A$$. 4. **Rewrite $$-8x - 12y$$:** Notice that $$-8x - 12y = -4(2x + 3y) = -4A$$. 5. **Substitute back:** The equation becomes $$A^2 - 4A + 16 = 0$$ 6. **Solve the quadratic equation for $$A$$:** Use the quadratic formula: $$A = \frac{4 \pm \sqrt{(-4)^2 - 4 \times 1 \times 16}}{2} = \frac{4 \pm \sqrt{16 - 64}}{2} = \frac{4 \pm \sqrt{-48}}{2}$$ 7. **Simplify the discriminant:** $$\sqrt{-48} = \sqrt{-1 \times 16 \times 3} = 4i\sqrt{3}$$ 8. **Final solutions:** $$A = \frac{4 \pm 4i\sqrt{3}}{2} = 2 \pm 2i\sqrt{3}$$ **Answer:** $$A = 2 + 2i\sqrt{3}$$ or $$A = 2 - 2i\sqrt{3}$$, which are complex numbers. This means there are no real values of $$A$$ satisfying the original equation; the solutions are complex.