Divide using synthetic division. \[ \frac{2 x^{5}-2 x^{4}+x^{3}-2 x^{2}-2 x-3}{x+2} \] \[ \frac{2 x^{5}-2 x^{4}+x^{3}-2 x^{2}-2 x-3}{x+2}=\square \] (Simplify your answer.)

Step 1 ：Given the polynomial division problem \(\frac{2 x^{5}-2 x^{4}+x^{3}-2 x^{2}-2 x-3}{x+2}\), we can solve it using synthetic division.

Step 2 ：First, we set up the synthetic division. The coefficients of the dividend (the polynomial we are dividing) are [2, -2, 1, -2, -2, -3]. The divisor is -2.

Step 3 ：We perform the synthetic division, which gives us the quotient [2, 2, 5, 8, 14, 25].

Step 4 ：This means that the quotient of the division is \(2x^4 + 2x^3 + 5x^2 + 8x + 14\) with a remainder of 25.

Step 5 ：Therefore, the division can be written as \(2x^4 + 2x^3 + 5x^2 + 8x + 14 + \frac{25}{x+2}\).

Step 6 ：So, the solution to the given polynomial division problem is \(\boxed{2x^4 + 2x^3 + 5x^2 + 8x + 14 + \frac{25}{x+2}}\).