Question

1. simplify: $(x^2 - 5x + 2)(x - 4)$

1. simplify: $(x^2 + 3x - 1)(x + 5)$

1. simplify: $(x^2 + 1)(x^2 - x + 1)$

1. simplify: $(2x^2 + 3x - 1)(x^2 - x + 2)$

Explanation:

Problem 17: Simplify \((x^2 - 5x + 2)(x - 4)\)

Step 1: Distribute \(x^2\)

Multiply \(x^2\) by each term in \((x - 4)\): \(x^2 \cdot x = x^3\), \(x^2 \cdot (-4) = -4x^2\)

Step 2: Distribute \(-5x\)

Multiply \(-5x\) by each term in \((x - 4)\): \(-5x \cdot x = -5x^2\), \(-5x \cdot (-4) = 20x\)

Step 3: Distribute \(2\)

Multiply \(2\) by each term in \((x - 4)\): \(2 \cdot x = 2x\), \(2 \cdot (-4) = -8\)

Step 4: Combine like terms

Combine the \(x^2\) terms: \(-4x^2 - 5x^2 = -9x^2\)
Combine the \(x\) terms: \(20x + 2x = 22x\)
The constant term is \(-8\)
So, putting it all together: \(x^3 - 9x^2 + 22x - 8\)

Step 1: Distribute \(x^2\)

Multiply \(x^2\) by each term in \((x + 5)\): \(x^2 \cdot x = x^3\), \(x^2 \cdot 5 = 5x^2\)

Step 2: Distribute \(3x\)

Multiply \(3x\) by each term in \((x + 5)\): \(3x \cdot x = 3x^2\), \(3x \cdot 5 = 15x\)

Step 3: Distribute \(-1\)

Multiply \(-1\) by each term in \((x + 5)\): \(-1 \cdot x = -x\), \(-1 \cdot 5 = -5\)

Step 4: Combine like terms

Combine the \(x^2\) terms: \(5x^2 + 3x^2 = 8x^2\)
Combine the \(x\) terms: \(15x - x = 14x\)
The constant term is \(-5\)
So, putting it all together: \(x^3 + 8x^2 + 14x - 5\)

Step 1: Distribute \(x^2\)

Multiply \(x^2\) by each term in \((x^2 - x + 1)\): \(x^2 \cdot x^2 = x^4\), \(x^2 \cdot (-x) = -x^3\), \(x^2 \cdot 1 = x^2\)

Step 2: Distribute \(1\)

Multiply \(1\) by each term in \((x^2 - x + 1)\): \(1 \cdot x^2 = x^2\), \(1 \cdot (-x) = -x\), \(1 \cdot 1 = 1\)

Step 3: Combine like terms

Combine the \(x^2\) terms: \(x^2 + x^2 = 2x^2\)
The \(x^3\) term is \(-x^3\)
The \(x^4\) term is \(x^4\)
The \(x\) term is \(-x\)
The constant term is \(1\)
So, putting it all together: \(x^4 - x^3 + 2x^2 - x + 1\)

Answer:

\(x^3 - 9x^2 + 22x - 8\)

Problem 18: Simplify \((x^2 + 3x - 1)(x + 5)\)