Divide. \[ \left(14 x+20 x^{3}+11+15 x^{2}\right) \div\left(5 x^{2}+2\right) \] Write your answer in the following form: Quotient \( +\frac{\text { Remainder }}{5 x^{2}+2} \). \( \frac{14 x+20 x^{3}+11+15 x^{2}}{5 x^{2}+2}=\square+\frac{\square}{5 x^{2}+2}+\square \times \square \)

To divide \( 14x + 20x^3 + 11 + 15x^2 \) by \( 5x^2 + 2 \), we first need to rearrange the polynomial in descending order of \( x \): \( 20x^3 + 15x^2 + 14x + 11 \) Now we perform polynomial long division: 1. Divide the first term of the dividend \( 20x^3 \) by the first term of the divisor \( 5x^2 \): - \( 20x^3 \div 5x^2 = 4x \) 2. Multiply \( 4x \) by the entire divisor \( 5x^2 + 2 \): - \( 4x(5x^2 + 2) = 20x^3 + 8x \) 3. Subtract this from the original polynomial: - \( (20x^3 + 15x^2 + 14x + 11) - (20x^3 + 8x) = 15x^2 + 6x + 11 \) 4. Now, repeat the process with the new polynomial \( 15x^2 + 6x + 11 \): - Divide \( 15x^2 \) by \( 5x^2 \): - \( 15x^2 \div 5x^2 = 3 \) 5. Multiply \( 3 \) by the whole divisor \( 5x^2 + 2 \): - \( 3(5x^2 + 2) = 15x^2 + 6 \) 6. Subtract this from the current polynomial: - \( (15x^2 + 6x + 11) - (15x^2 + 6) = 6x + 5 \) At this point, \( 6x + 5 \) is the remainder. Since the degree of the remainder is less than the degree of the divisor, we can stop here. Combining all parts, we have: \[ \left(14 x + 20 x^{3} + 11 + 15 x^{2}\right) \div \left(5 x^{2} + 2\right) = 4x + 3 + \frac{6x + 5}{5x^2 + 2} \] Therefore, in the requested form, our final answer is: \[ \frac{14 x + 20 x^{3} + 11 + 15 x^{2}}{5 x^{2} + 2} = 4x + 3 + \frac{6x + 5}{5 x^{2} + 2} \]