To start, we are given the polynomial dividend of -6x^4 + 7x^2 + 5x + 18 and the polynomial divisor of -2x^2 - 3. We want to divide the dividend by the divisor.
Let's organize it using polynomial long division:
- We first divide the leading term of the dividend, -6x^4, by the leading term of the divisor, -2x^2. This gives us 3x^2.
- Multiply all terms in the divisor by 3x^2 and subtract that from our original dividend. This gives us -( -6x^4 + 6x^2 ) + 7x^2 + 5x + 18 = -8x^2 + 5x + 18 as the new dividend.
- Repeat the process by dividing -8x^2 (leading term of the new dividend) by -2x^2 (leading term of the divisor). This results in additional 4 to our quotient, and -8x^2 + 8x^2 = 0 leaves us with remainder of 5x + 18 after subtraction.
- At this point, we can notice that the degree of what we have left (5x + 18) is less than the degree of our divisor (-2x^2 -3), so we can stop here.
Our quotient is 3x^2 - 4 and our remainder is 5x + 18.
So, the result of the division is given: Quotient + (Remainder/divisor), in other words: (3x^2 - 4) + (5x + 18)/(-2x^2 - 3).
Note: Just a reminder that polynomial division is exactly the same procedure as the long division you would have learned in primary school, but with extra repeated steps for each term in the dividend.