Divide. \[ \left(2 x^{5}+3 x^{3}+2 x^{2}+12 x+4\right) \div\left(x^{2}+2\right) \]

Step 1 ：Let's solve the given polynomial division problem: \(\left(2 x^{5}+3 x^{3}+2 x^{2}+12 x+4\right) \div\left(x^{2}+2\right)\)

Step 2 ：We can solve this by using the long division method for polynomials.

Step 3 ：Performing the division, we get the quotient and the remainder.

Step 4 ：The quotient is \(2 x^{3}-4 x^{2}+8 x+16\) and the remainder is \(32 x+36\)

Step 5 ：\(\boxed{\text{Quotient} = 2 x^{3}-4 x^{2}+8 x+16, \text{Remainder} = 32 x+36}\)