To find the quotient of the polynomial division of 6x^4 + 15x^3 + 10x^2 + 10x + 4 by 3x^2 + 2, we can use polynomial long division.
1. **Setup the division**: Write the dividend (the polynomial you are dividing) and the divisor (the polynomial you are dividing by).
2. **Divide the leading terms**: Divide the leading term of the dividend (6x^4) by the leading term of the divisor (3x^2), which gives you 2x^2. This will be the first term of our quotient.
3. **Multiply and subtract**: Multiply the entire divisor (3x^2 + 2) by 2x^2, which gives us 6x^4 + 4x^2. Now, subtract this from the original polynomial.
4. After subtraction:
(6x^4 + 15x^3 + 10x^2 + 10x + 4) – (6x^4 + 4x^2) = 15x^3 + 6x^2 + 10x + 4.
5. **Repeat the process**: Now, repeat the steps with the new polynomial (15x^3 + 6x^2 + 10x + 4). Divide the leading term (15x^3) by the leading term of the divisor (3x^2), giving us 5x. Multiply the entire divisor by 5x and subtract again.
6. Continue until the degree of the new polynomial is less than the degree of the divisor. Eventually, you will reach a polynomial or a constant remainder.
The final quotient will combine all the terms you found during the division steps. Thus, the quotient of your polynomial division can be expressed as:
2x^2 + 5x + R, where R is the remainder obtained in the last step.
This method provides a systematic approach to finding the quotient of polynomial expressions, ensuring you carefully address each term.