Algebra 2 worksheet featuring polynomial division problems using long division.

Since the user has set both `ExplainAnswer` and `TeachTopic` to false, I will proceed directly to solving the problems as requested, verifying my calculations before providing the final answers.

Here are the solutions for the polynomial long division problems shown in the image.

Problem 1: $(k^3 + 8k^2 + 10k + 21) \div (k + 7)$

1. Divide: $k^3 \div k = k^2$
2. Multiply: $k^2(k + 7) = k^3 + 7k^2$
3. Subtract: $(k^3 + 8k^2) - (k^3 + 7k^2) = k^2$
4. Bring down: $10k$
5. Divide: $k^2 \div k = k$
6. Multiply: $k(k + 7) = k^2 + 7k$
7. Subtract: $(k^2 + 10k) - (k^2 + 7k) = 3k$
8. Bring down: $21$
9. Divide: $3k \div k = 3$
10. Multiply: $3(k + 7) = 3k + 21$
11. Subtract: $(3k + 21) - (3k + 21) = 0$
Result: $k^2 + k + 3$

Problem 2: $(n^4 - 17n^3 + 81n^2 - 65n - 56) \div (n - 8)$

1. Divide: $n^4 \div n = n^3$
2. Multiply: $n^3(n - 8) = n^4 - 8n^3$
3. Subtract: $-17n^3 - (-8n^3) = -9n^3$
4. Bring down: $81n^2$
5. Divide: $-9n^3 \div n = -9n^2$
6. Multiply: $-9n^2(n - 8) = -9n^3 + 72n^2$
7. Subtract: $81n^2 - 72n^2 = 9n^2$
8. Bring down: $-65n$
9. Divide: $9n^2 \div n = 9n$
10. Multiply: $9n(n - 8) = 9n^2 - 72n$
11. Subtract: $-65n - (-72n) = 7n$
12. Bring down: $-56$
13. Divide: $7n \div n = 7$
14. Multiply: $7(n - 8) = 7n - 56$
15. Subtract: $-56 - (-56) = 0$
Result: $n^3 - 9n^2 + 9n + 7$

Problem 3: $(7n^4 - 68n^3 + 46n^2 - 7n - 18) \div (n - 9)$

1. Divide: $7n^4 \div n = 7n^3$
2. Multiply: $7n^3(n - 9) = 7n^4 - 63n^3$
3. Subtract: $-68n^3 - (-63n^3) = -5n^3$
4. Bring down: $46n^2$
5. Divide: $-5n^3 \div n = -5n^2$
6. Multiply: $-5n^2(n - 9) = -5n^3 + 45n^2$
7. Subtract: $46n^2 - 45n^2 = n^2$
8. Bring down: $-7n$
9. Divide: $n^2 \div n = n$
10. Multiply: $n(n - 9) = n^2 - 9n$
11. Subtract: $-7n - (-9n) = 2n$
12. Bring down: $-18$
13. Divide: $2n \div n = 2$
14. Multiply: $2(n - 9) = 2n - 18$
15. Subtract: $-18 - (-18) = 0$
Result: $7n^3 - 5n^2 + n + 2$

Problem 4: $(5k^4 + 14k^3 + 12k^2 + 14k + 12) \div (k + 2)$

1. Divide: $5k^4 \div k = 5k^3$
2. Multiply: $5k^3(k + 2) = 5k^4 + 10k^3$
3. Subtract: $14k^3 - 10k^3 = 4k^3$
4. Bring down: $12k^2$
5. Divide: $4k^3 \div k = 4k^2$
6. Multiply: $4k^2(k + 2) = 4k^3 + 8k^2$
7. Subtract: $12k^2 - 8k^2 = 4k^2$
8. Bring down: $14k$
9. Divide: $4k^2 \div k = 4k$
10. Multiply: $4k(k + 2) = 4k^2 + 8k$
11. Subtract: $14k - 8k = 6k$
12. Bring down: $12$
13. Divide: $6k \div k = 6$
14. Multiply: $6(k + 2) = 6k + 12$
15. Subtract: $12 - 12 = 0$
Result: $5k^3 + 4k^2 + 4k + 6$

Problem 5: $(-8x^3 + 40x^2 - 37x + 30) \div (x - 4)$

1. Divide: $-8x^3 \div x = -8x^2$
2. Multiply: $-8x^2(x - 4) = -8x^3 + 32x^2$
3. Subtract: $40x^2 - 32x^2 = 8x^2$
4. Bring down: $-37x$
5. Divide: $8x^2 \div x = 8x$
6. Multiply: $8x(x - 4) = 8x^2 - 32x$
7. Subtract: $-37x - (-32x) = -5x$
8. Bring down: $30$
9. Divide: $-5x \div x = -5$
10. Multiply: $-5(x - 4) = -5x + 20$
11. Subtract: $30 - 20 = 10$
Result: $-8x^2 + 8x - 5$ with a remainder of $10$

Problem 6: $(x^4 + 2x^3 - 87x^2 - 68x + 13) \div (x + 10)$

1. Divide: $x^4 \div x = x^3$
2. Multiply: $x^3(x + 10) = x^4 + 10x^3$
3. Subtract: $2x^3 - 10x^3 = -8x^3$
4. Bring down: $-87x^2$
5. Divide: $-8x^3 \div x = -8x^2$
6. Multiply: $-8x^2(x + 10) = -8x^3 - 80x^2$
7. Subtract: $-87x^2 - (-80x^2) = -7x^2$
8. Bring down: $-68x$
9. Divide: $-7x^2 \div x = -7x$
10. Multiply: $-7x(x + 10) = -7x^2 - 70x$
11. Subtract: $-68x - (-70x) = 2x$
12. Bring down: $13$
13. Divide: $2x \div x = 2$
14. Multiply: $2(x + 10) = 2x + 20$
15. Subtract: $13 - 20 = -7$
Result: $x^3 - 8x^2 - 7x + 2$ with a remainder of $-7$

Problem 7: $(12v^3 + 38v^2 - 16v - 20) \div (6v - 5)$

1. Divide: $12v^3 \div 6v = 2v^2$
2. Multiply: $2v^2(6v - 5) = 12v^3 - 10v^2$
3. Subtract: $38v^2 - (-10v^2) = 48v^2$
4. Bring down: $-16v$
5. Divide: $48v^2 \div 6v = 8v$
6. Multiply: $8v(6v - 5) = 48v^2 - 40v$
7. Subtract: $-16v - (-40v) = 24v$
8. Bring down: $-20$
9. Divide: $24v \div 6v = 4$
10. Multiply: $4(6v - 5) = 24v - 20$
11. Subtract: $-20 - (-20) = 0$
Result: $2v^2 + 8v + 4$

Problem 8: $(4x^4 - 15x^3 - 28x^2 + 6x + 3) \div (4x + 1)$

1. Divide: $4x^4 \div 4x = x^3$
2. Multiply: $x^3(4x + 1) = 4x^4 + x^3$
3. Subtract: $-15x^3 - x^3 = -16x^3$
4. Bring down: $-28x^2$
5. Divide: $-16x^3 \div 4x = -4x^2$
6. Multiply: $-4x^2(4x + 1) = -16x^3 - 4x^2$
7. Subtract: $-28x^2 - (-4x^2) = -24x^2$
8. Bring down: $6x$
9. Divide: $-24x^2 \div 4x = -6x$
10. Multiply: $-6x(4x + 1) = -24x^2 - 6x$
11. Subtract: $6x - (-6x) = 12x$
12. Bring down: $3$
13. Divide: $12x \div 4x = 3$
14. Multiply: $3(4x + 1) = 12x + 3$
15. Subtract: $3 - 3 = 0$
Result: $x^3 - 4x^2 - 6x + 3$

Final Answer:
1) $k^2 + k + 3$
2) $n^3 - 9n^2 + 9n + 7$
3) $7n^3 - 5n^2 + n + 2$
4) $5k^3 + 4k^2 + 4k + 6$
5) $-8x^2 + 8x - 5$ R $10$
6) $x^3 - 8x^2 - 7x + 2$ R $-7$
7) $2v^2 + 8v + 4$
8) $x^3 - 4x^2 - 6x + 3$