To solve this problem, we need to perform synthetic division on the polynomial 3x^2 + 10x - 8 with the divisor x + 4.
Step 1: Write down the coefficients of the polynomial in order. The polynomial is 3x^2 + 10x - 8, so the coefficients are 3, 10, and -8.
Step 2: Write down the "zero" of the divisor x + 4. The zero is the number that makes the divisor equal to zero. In this case, it's -4.
Step 3: Set up the synthetic division table. You can write -4 (the zero of the divisor) on the left, then draw a bar, then write the coefficients (3, 10, -8) to the right of the bar:
-4 | 3 10 -8
Step 4: Bring down the leading coefficient (3 in this case) in the polynomial to the bottom row.
-4 | 3 10 -8
| __
3
Step 5: Multiply the number you just brought down (3) by the zero (-4), and write the product under the second number of the top row (10).
-4 | 3 10 -8
| -12
3
Step 6: Add together the numbers in the second column, and write the sum in the second position of the bottom row.
-4 | 3 10 -8
| -12
3 -2
Step 7: Repeat steps 5 and 6 until you've filled the bottom row.
-4 | 3 10 -8
| -12 8
3 -2 0
After doing this synthetic division, the numbers on the bottom row form the coefficients of the quotient polynomial. The last number on the bottom row is the remainder (if it's not zero). In this case, the quotient is the polynomial 3x - 2 and the remainder is zero. So, 3x^2 + 10x - 8 divided by x + 4 gives a quotient of 3x - 2 and a remainder of 0.