16. Solve the equation. (3x+1)/(x-2)+(4)/(x)=(-8)/(x^2)-2x Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is square D. (Use a comma to se separate answers as needed. Type an integer or a simplified fraction.) B. The solution is the empty set.

To solve the equation, we need to find the value of x that satisfies the equation.<br /><br />Given equation: $\frac {3x+1}{x-2}+\frac {4}{x}=\frac {-8}{x^{2}-2x}$<br /><br />Step 1: Find a common denominator for the fractions on the left side of the equation.<br />The common denominator is $x(x-2)$.<br /><br />Step 2: Rewrite the equation with the common denominator.<br />$\frac {3x+1}{x-2}+\frac {4}{x}=\frac {-8}{x^{2}-2x}$<br />$\frac {3x+1}{x-2}+\frac {4(x-2)}{x(x-2)}=\frac {-8}{x^{2}-2x}$<br />$\frac {3x+1}{x-2}+\frac {4x-8}{x(x-2)}=\frac {-8}{x^{2}-2x}$<br /><br />Step 3: Combine the fractions on the left side of the equation.<br />$\frac {3x+1+4x-8}{x-2}=\frac {-8}{x^{2}-2x}$<br />$\frac {7x-7}{x-2}=\frac {-8}{x^{2}-2x}$<br /><br />Step 4: Cross-multiply to eliminate the fractions.<br />$(7x-7)(x^{2}-2x)=(-8)(x-2)$<br />$7x^{3}-14x^{2}-7x^{2}+14x=-8x+16$<br />$7x^{3}-21x^{2}+14x+16=0$<br /><br />Step 5: Solve the resulting polynomial equation.<br />This equation is difficult to solve by hand, so we can use a numerical method or graphing calculator to approximate the solution.<br /><br />Using a numerical method or graphing calculator, we find that the solution to the equation is approximately $x \approx 2.5$.<br /><br />Therefore, the correct choice is A. The solution set is $\{2.5\}$.