Step 1: Group like terms: (3x² + 2x²) + (5x - 3x) + (-2 + 7)

Step 2: Combine coefficients: 5x² + 2x + 5

Answer: 5x² + 2x + 5

Distribute each term in the first polynomial:

$x(x^2 + 3x + 2) + 5(x^2 + 3x + 2)$

$= x^3 + 3x^2 + 2x + 5x^2 + 15x + 10$

Combine like terms: $= x^3 + 8x^2 + 17x + 10$

Answer: $x^3 + 8x^2 + 17x + 10$

Step 1: Distribute 2x to each term in the second polynomial: 2x(x²) + 2x(-4x) + 2x(5) = 2x³ - 8x² + 10x

Step 2: Distribute 3 to each term: 3(x²) + 3(-4x) + 3(5) = 3x² - 12x + 15

Step 3: Combine all terms: 2x³ - 8x² + 10x + 3x² - 12x + 15

Step 4: Combine like terms: 2x³ + (-8x² + 3x²) + (10x - 12x) + 15 2x³ - 5x² - 2x + 15

Answer: 2x³ - 5x² - 2x + 15

By the Remainder Theorem, the remainder when dividing by $(x - 2)$ is $P(2)$.

Evaluate $P(2)$: $P(2) = (2)^3 - 4(2)^2 + 6(2) - 2$ $= 8 - 16 + 12 - 2$ $= 2$

Answer: Remainder = $2$

Step 1: Distribute the negative sign: 4x³ + 2x² - 7x + 1 - 2x³ + 3x² - 5x + 4

Step 2: Group like terms: (4x³ - 2x³) + (2x² + 3x²) + (-7x - 5x) + (1 + 4)

Step 3: Combine: 2x³ + 5x² - 12x + 5

Step 4: Verify by plugging in x = 1: Original: (4 + 2 - 7 + 1) - (2 - 3 + 5 - 4) = 0 - 0 = 0 Result: 2 + 5 - 12 + 5 = 0 ✓

Answer: 2x³ + 5x² - 12x + 5

Step 1: Multiply the first two factors: (x + 2)(x - 3) = x² - 3x + 2x - 6 = x² - x - 6

Step 2: Multiply the result by the third factor: (x² - x - 6)(x + 4)

Step 3: Distribute x: x(x²) + x(-x) + x(-6) = x³ - x² - 6x

Step 4: Distribute 4: 4(x²) + 4(-x) + 4(-6) = 4x² - 4x - 24

Step 5: Combine all terms: x³ - x² - 6x + 4x² - 4x - 24 = x³ + 3x² - 10x - 24

Step 6: Verify by checking the constant term: Product of constants: 2 × (-3) × 4 = -24 ✓

Answer: x³ + 3x² - 10x - 24

Set up long division:

Step 1: $2x^3 \div x = 2x^2$ Multiply: $2x^2(x + 2) = 2x^3 + 4x^2$ Subtract: $(2x^3 + 3x^2) - (2x^3 + 4x^2) = -x^2$

Step 2: $-x^2 \div x = -x$ Multiply: $-x(x + 2) = -x^2 - 2x$ Subtract: $(-x^2 - 5x) - (-x^2 - 2x) = -3x$

Step 3: $-3x \div x = -3$ Multiply: $-3(x + 2) = -3x - 6$ Subtract: $(-3x + 1) - (-3x - 6) = 7$

Answer: $2x^2 - x - 3 + \frac{7}{x + 2}$

Step 1: Use the binomial expansion formula: (a + b)³ = a³ + 3a²b + 3ab² + b³

Step 2: Identify a = 2x and b = -1: (2x)³ + 3(2x)²(-1) + 3(2x)(-1)² + (-1)³

Step 3: Calculate each term: (2x)³ = 8x³ 3(2x)²(-1) = 3(4x²)(-1) = -12x² 3(2x)(-1)² = 3(2x)(1) = 6x (-1)³ = -1

Step 4: Combine: 8x³ - 12x² + 6x - 1

Step 5: Alternative method - multiply step by step: (2x - 1)² = 4x² - 4x + 1 (4x² - 4x + 1)(2x - 1) = 8x³ - 4x² - 8x² + 4x + 2x - 1 = 8x³ - 12x² + 6x - 1 ✓

Answer: 8x³ - 12x² + 6x - 1