Use synthetic division to determine the remainder when 5x4 + 7x3 + x2 − 18 is divided by x + 2.

Sure, here's how you can use synthetic division to find the remainder when 5x^4 + 7x^3 + x^2 - 18 is divided by x + 2.

Step 1: Write down the coefficients of the dividend (5x^4 + 7x^3 + x^2 - 18), which are 5, 7, 1, 0 (for x), and -18.

Step 2: Write down the zero of the divisor (x + 2). The zero is the solution to the equation x + 2 = 0, which is x = -2.

Step 3: Set up the synthetic division. Draw a line, write -2 on the left, and the coefficients from step 1 on the right:

        -2 | 5  7  1  0  -18

Step 4: Bring down the first coefficient (5) to start the division:

        -2 | 5  7  1  0  -18
            -----------------
              5

Step 5: Multiply the number you just wrote (5) by -2 (the zero of the divisor), and write the result (-10) under the next coefficient (7):

        -2 | 5  7  1  0  -18
            -   10
            -----------------
              5  -3

Step 6: Add the numbers in the second column (7 and -10), and write the result (-3) under the line:

        -2 | 5  7  1  0  -18
            -  10
            -----------------
              5  -3

Step 7: Repeat steps 5 and 6 for the rest of the coefficients. Multiply -3 by -2 to get 6, write it under 1, add to get 7. Multiply 7 by -2 to get -14, write it under 0, add to get -14. Multiply -14 by -2 to get 28, write it under -18, add to get 10.

        -2 | 5  7  1  0  -18
            -  10  6 -14  28
            -----------------
              5  -3  7 -14  10

Step 8: The numbers under the line represent the coefficients of the quotient and the remainder. The last number (10) is the remainder. So, the remainder when 5x^4 + 7x^3 + x^2 - 18 is divided by x + 2 is 10.