Use synthetic division to find the quotient and remainder: (2x^5-7x^3-5x^2+1)/(x-2)

This is what I got:
Quotient: 2x^4+4x^3+x^2-3x-6
Remainder: -11

Do i just leave the quotient like this: 2x^4+4x^3+x^2-3x-6?
Do I need to factor it down more?

please help and thank you

Your division is correct

Since you have a remainder, the expression cannot be properly factored
the best you can so is say ...

(2x^5-7x^3-5x^2+1)/(x-2) = 2x^4+4x^3+x^2-3x-6 - 11/(x-2)

To use synthetic division, follow these steps:

1. Write the coefficients of the dividend in descending order of the powers of x, adding placeholders for missing terms. In this case, the dividend is 2x^5 - 7x^3 - 5x^2 + 1, so it can be written as 2x^5 + 0x^4 - 7x^3 - 5x^2 + 0x + 1.

2. Write the divisor to the left of the dividend. In this case, the divisor is x - 2.

x - 2 | 2 0 -7 -5 0 1

3. Bring down the first coefficient, which is 2.

x - 2 | 2 0 -7 -5 0 1
2

4. Multiply the divisor (x - 2) by the first term of the quotient (2) and write the result below the second coefficient.

x - 2 | 2 0 -7 -5 0 1
2x - 4

5. Add the second and third columns together and write the result below the third coefficient.

x - 2 | 2 0 -7 -5 0 1
2x - 4
____________
2 2 -11

6. Repeat steps 4 and 5 using the new polynomial (2x - 4).

x - 2 | 2 0 -7 -5 0 1
2x - 4
____________
2 2 -11
2 4
____________
2 4 -7

7. Continue repeating steps 4 and 5 until you have gone through all the coefficients. The last row represents the remainder.

The final result of synthetic division is:
Quotient: 2x^4 + 4x^3 - 7
Remainder: 2x^2 + 4x - 7

You don't need to factor the quotient further unless specifically required. However, if you want to factor the quotient, you can try factoring it using techniques such as factoring by grouping or factoring using the quadratic formula.