1. Problem: Find the quotient and remainder when dividing the polynomial $$2x^3 + x^2 - x - 4$$ by $$x - 2$$. 2. Use polynomial long division: Divide the leading term $$2x^3$$ by $$x$$ to get $$2x^2$$. 3. Multiply $$2x^2 (x - 2) = 2x^3 - 4x^2$$. 4. Subtract from the original polynomial: $$(2x^3 + x^2) - (2x^3 - 4x^2) = 0 + 5x^2$$. 5. Bring down remaining terms: $$5x^2 - x - 4$$. 6. Divide leading term $$5x^2$$ by $$x$$ to get $$5x$$. 7. Multiply $$5x(x - 2) = 5x^2 - 10x$$. 8. Subtract: $$(5x^2 - x) - (5x^2 - 10x) = 0 + 9x$$. 9. Bring down $$-4$$, resulting in $$9x - 4$$. 10. Divide leading term $$9x$$ by $$x$$ to get $$9$$. 11. Multiply $$9(x - 2) = 9x - 18$$. 12. Subtract: $$(9x - 4) - (9x - 18) = 0 + 14$$. 13. Quotient is $$2x^2 + 5x + 9$$, remainder is $$14$$. -- 1. Problem 8(a): Find quotient and remainder of $$x^3 - 5x^2 + 2x - 10$$ divided by $$x - 5$$. 2. Divide leading term $$x^3$$ by $$x$$ to get $$x^2$$. 3. Multiply $$x^2(x - 5) = x^3 - 5x^2$$. 4. Subtract: $$(x^3 - 5x^2) - (x^3 - 5x^2) = 0$$. 5. Bring down $$2x - 10$$. 6. Divide $$2x$$ by $$x$$ to get $$2$$. 7. Multiply $$2(x - 5) = 2x - 10$$. 8. Subtract: $$(2x - 10) - (2x - 10) = 0$$. 9. Quotient is $$x^2 + 2$$, remainder is $$0$$. 10. Problem 8(b): Since remainder is 0, $$x - 5$$ is a factor. 11. Then $$p(x) = (x - 5)(x^2 + 2)$$. -- 1. Problem 9(a): Divide $$x^3 + x^2 + 2x + 1$$ by $$x - 2$$. 2. Divide $$x^3$$ by $$x$$ to get $$x^2$$. 3. Multiply $$x^2(x - 2) = x^3 - 2x^2$$. 4. Subtract: $$(x^3 + x^2) - (x^3 - 2x^2) = 3x^2$$. 5. Bring down $$+ 2x$$. 6. Divide $$3x^2$$ by $$x$$ to get $$3x$$. 7. Multiply $$3x(x - 2) = 3x^2 - 6x$$. 8. Subtract: $$(3x^2 + 2x) - (3x^2 - 6x) = 8x$$. 9. Bring down $$+ 1$$. 10. Divide $$8x$$ by $$x$$ to get $$8$$. 11. Multiply $$8(x - 2) = 8x - 16$$. 12. Subtract: $$(8x + 1) - (8x - 16) = 17$$. 13. Quotient is $$x^2 + 3x + 8$$, remainder $$17$$. -- 1. Problem 9(b): Divide $$4x^3 + 2x^2 + 2x - 3$$ by $$x - 2$$. 2. Divide $$4x^3$$ by $$x$$ to get $$4x^2$$. 3. Multiply $$4x^2(x - 2) = 4x^3 - 8x^2$$. 4. Subtract: $$(4x^3 + 2x^2) - (4x^3 - 8x^2) = 10x^2$$. 5. Bring down $$+ 2x$$. 6. Divide $$10x^2$$ by $$x$$ to get $$10x$$. 7. Multiply $$10x(x - 2) = 10x^2 - 20x$$. 8. Subtract: $$(10x^2 + 2x) - (10x^2 - 20x) = 22x$$. 9. Bring down $$- 3$$. 10. Divide $$22x$$ by $$x$$ to get $$22$$. 11. Multiply $$22(x - 2) = 22x - 44$$. 12. Subtract: $$(22x - 3) - (22x - 44) = 41$$. 13. Quotient is $$4x^2 + 10x + 22$$, remainder $$41$$. -- 1. Problem 10(a): Divide $$3x^3 - 2x - 2$$ by $$x + 2$$. 2. Divide $$3x^3$$ by $$x$$ to get $$3x^2$$. 3. Multiply $$3x^2(x + 2) = 3x^3 + 6x^2$$. 4. Subtract: $$(3x^3 + 0x^2) - (3x^3 + 6x^2) = -6x^2$$. 5. Bring down $$-2x$$. 6. Divide $$-6x^2$$ by $$x$$ to get $$-6x$$. 7. Multiply $$-6x(x + 2) = -6x^2 - 12x$$. 8. Subtract: $$(-6x^2 - 2x) - (-6x^2 - 12x) = 10x$$. 9. Bring down $$-2$$. 10. Divide $$10x$$ by $$x$$ to get $$10$$. 11. Multiply $$10(x + 2) = 10x + 20$$. 12. Subtract: $$(10x - 2) - (10x + 20) = -22$$. 13. Quotient: $$3x^2 - 6x + 10$$, remainder $$-22$$. -- 1. Problem 10(b): Divide $$-2x^3 - 3x^2 + 8x + 5$$ by $$2x - 1$$. 2. Divide leading term $$-2x^3$$ by $$2x$$ to get $$-x^2$$. 3. Multiply $$-x^2(2x - 1) = -2x^3 + x^2$$. 4. Subtract: $$(-2x^3 - 3x^2) - (-2x^3 + x^2) = -4x^2$$. 5. Bring down $$+ 8x$$. 6. Divide $$-4x^2$$ by $$2x$$ to get $$-2x$$. 7. Multiply $$-2x(2x - 1) = -4x^2 + 2x$$. 8. Subtract: $$(-4x^2 + 8x) - (-4x^2 + 2x) = 6x$$. 9. Bring down $$+ 5$$. 10. Divide $$6x$$ by $$2x$$ to get $$3$$. 11. Multiply $$3(2x - 1) = 6x - 3$$. 12. Subtract: $$(6x + 5) - (6x - 3) = 8$$. 13. Quotient is $$-x^2 - 2x + 3$$, remainder $$8$$. -- Total distinct problems answered: 8 (Problems 7, 8a, 8b, 9a, 9b, 10a, 10b).