Question

multiply and write in standard form.
1
$(8x + 5)(x^2 - 3x - 1)$
2
$(x - 6)(6x^2 + 3x + 5)$
3
$(2x + 4)(8x^2 - 2x + 10)$

Explanation:

Problem 1: \((8x + 5)(x^2 - 3x - 1)\)

Step 1: Distribute \(8x\) and \(5\)

Multiply \(8x\) by each term in \((x^2 - 3x - 1)\) and \(5\) by each term in \((x^2 - 3x - 1)\):
\(8x \cdot x^2 + 8x \cdot (-3x) + 8x \cdot (-1) + 5 \cdot x^2 + 5 \cdot (-3x) + 5 \cdot (-1)\)
\(= 8x^3 - 24x^2 - 8x + 5x^2 - 15x - 5\)

Step 2: Combine like terms

Combine the \(x^2\) terms and the \(x\) terms:
\(8x^3 + (-24x^2 + 5x^2) + (-8x - 15x) - 5\)
\(= 8x^3 - 19x^2 - 23x - 5\)

Step 1: Distribute \(x\) and \(-6\)

Multiply \(x\) by each term in \((6x^2 + 3x + 5)\) and \(-6\) by each term in \((6x^2 + 3x + 5)\):
\(x \cdot 6x^2 + x \cdot 3x + x \cdot 5 + (-6) \cdot 6x^2 + (-6) \cdot 3x + (-6) \cdot 5\)
\(= 6x^3 + 3x^2 + 5x - 36x^2 - 18x - 30\)

Step 2: Combine like terms

Combine the \(x^2\) terms and the \(x\) terms:
\(6x^3 + (3x^2 - 36x^2) + (5x - 18x) - 30\)
\(= 6x^3 - 33x^2 - 13x - 30\)

Step 1: Distribute \(2x\) and \(4\)

Multiply \(2x\) by each term in \((8x^2 - 2x + 10)\) and \(4\) by each term in \((8x^2 - 2x + 10)\):
\(2x \cdot 8x^2 + 2x \cdot (-2x) + 2x \cdot 10 + 4 \cdot 8x^2 + 4 \cdot (-2x) + 4 \cdot 10\)
\(= 16x^3 - 4x^2 + 20x + 32x^2 - 8x + 40\)

Step 2: Combine like terms

Combine the \(x^2\) terms and the \(x\) terms:
\(16x^3 + (-4x^2 + 32x^2) + (20x - 8x) + 40\)
\(= 16x^3 + 28x^2 + 12x + 40\)

Answer:

\(8x^3 - 19x^2 - 23x - 5\)

Problem 2: \((x - 6)(6x^2 + 3x + 5)\)