The question requires us to perform a polynomial division of the function 3x^3 + 17x^2 - 30x - 14 by the function x + 7. Let's follow the steps to achieve this.
Step 1: Write the dividend and divisor
We will write the problem in the form of long division, with 3x^3 + 17x^2 - 30x - 14 as the dividend (equivalent of numerator in a fraction) and x + 7 as the divisor (equivalent of denominator in a fraction).
Step 2: Divide the first term of the dividend by the first term of the divisor
We divide the first term in the dividend, 3x^3, by the first term in the divisor, x.
This yields a quotient of 3x^2.
Step 3: Multiply the divisor by the quotient and subtract the result from the dividend
The divisor (x + 7) is multiplied by the new part of our answer, 3x^2, to get 3x^3 + 21x^2. We subtract this new expression from our original dividend.
This transforms our original dividend of 3x^3 + 17x^2 - 30x - 14 down to -4x^2 - 30x - 14.
Step 4: Repeat Step 2 and 3
We repeat steps 2 and 3, now taking -4x^2 to divide by x to get -4x, our new term for the quotient. We multiply through by the divisor to get -4x^2 - 28x, and subtract to get -2x - 14.
We go back again and divide -2x by x to get a quotient of -2. Multiplying through and subtracting results in no remainder.
The process concludes here as we have reached a degree of zero and cannot continue the division process.
So, the polynomial 3x^3 + 17x^2 - 30x - 14 divided by the polynomial x + 7 is 3x^2 - 4x - 2 (because we have no remainder).