Question

1. (4x²+3x+2)(3x²+2x-1)
2. (x-2)(3x+1)(x+3)

Explanation:

Problem 9: \((4x^2 + 3x + 2)(3x^2 + 2x - 1)\)

Step 1: Apply the distributive property (FOIL for polynomials)

Multiply each term in the first polynomial by each term in the second polynomial:
\[

$$\begin{align*} &4x^2 \cdot 3x^2 + 4x^2 \cdot 2x + 4x^2 \cdot (-1) + 3x \cdot 3x^2 + 3x \cdot 2x + 3x \cdot (-1) + 2 \cdot 3x^2 + 2 \cdot 2x + 2 \cdot (-1)\\ &= 12x^4 + 8x^3 - 4x^2 + 9x^3 + 6x^2 - 3x + 6x^2 + 4x - 2 \end{align*}$$

\]

Step 2: Combine like terms

• For \(x^4\): \(12x^4\)
• For \(x^3\): \(8x^3 + 9x^3 = 17x^3\)
• For \(x^2\): \(-4x^2 + 6x^2 + 6x^2 = 8x^2\)
• For \(x\): \(-3x + 4x = x\)
• Constant term: \(-2\)

Putting it all together: \(12x^4 + 17x^3 + 8x^2 + x - 2\)

Step 1: Multiply the first two binomials \((x - 2)(3x + 1)\)

Using the distributive property:
\[

$$\begin{align*} &x \cdot 3x + x \cdot 1 - 2 \cdot 3x - 2 \cdot 1\\ &= 3x^2 + x - 6x - 2\\ &= 3x^2 - 5x - 2 \end{align*}$$

\]

Step 2: Multiply the result by \((x + 3)\)

\[

$$\begin{align*} &(3x^2 - 5x - 2)(x + 3)\\ &= 3x^2 \cdot x + 3x^2 \cdot 3 - 5x \cdot x - 5x \cdot 3 - 2 \cdot x - 2 \cdot 3\\ &= 3x^3 + 9x^2 - 5x^2 - 15x - 2x - 6 \end{align*}$$

\]

Step 3: Combine like terms

• For \(x^3\): \(3x^3\)
• For \(x^2\): \(9x^2 - 5x^2 = 4x^2\)
• For \(x\): \(-15x - 2x = -17x\)
• Constant term: \(-6\)

Putting it all together: \(3x^3 + 4x^2 - 17x - 6\)

Answer:

\(12x^4 + 17x^3 + 8x^2 + x - 2\)

Problem 10: \((x - 2)(3x + 1)(x + 3)\)