Let's start solving the four multiplication problems in "LOCK #2" one by one, and then use the "Decoder" to match the answers.

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

Let's expand this expression using the distributive property:

$(x^2 - 4x - 1)(x^2 + x + 1)$

Expanding step by step:

• $x^2 \times (x^2 + x + 1) = x^4 + x^3 + x^2$
• $-4x \times (x^2 + x + 1) = -4x^3 - 4x^2 - 4x$
• $-1 \times (x^2 + x + 1) = -x^2 - x - 1$

Now combine like terms:

$x^4 + x^3 + x^2 - 4x^3 - 4x^2 - 4x - x^2 - x - 1$

Simplifying:

$x^4 - 3x^3 - 4x^2 - 5x - 1$

So, the result is:

$x^4 - 3x^3 - 4x^2 - 5x - 1 \quad \text{(Matches with option A in the decoder)}$

2. Multiply $(3x - 1)^2$:

This is a perfect square:

$(3x - 1)^2 = (3x - 1)(3x - 1)$

Expand it:

$3x \times 3x = 9x^2$ $3x \times (-1) = -3x$ $-1 \times 3x = -3x$ $-1 \times -1 = 1$

Combine like terms:

$9x^2 - 6x + 1$

So, the result is:

$9x^2 - 6x + 1 \quad \text{(Matches with option N in the decoder)}$

3. Multiply $2x^2(x + 5)$:

Distribute $2x^2$ to both terms inside the parentheses:

$2x^2 \times x = 2x^3$ $2x^2 \times 5 = 10x^2$

So, the result is:

$2x^3 + 10x^2 \quad \text{(Matches with option D in the decoder)}$

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

Expand this expression using the distributive property:

$(x - 1)(x^2 + x + 1)$

Expanding step by step:

• $x \times (x^2 + x + 1) = x^3 + x^2 + x$
• $-1 \times (x^2 + x + 1) = -x^2 - x - 1$

Now combine like terms:

$x^3 + x^2 + x - x^2 - x - 1$

Simplifying:

$x^3 - 1$

So, the result is:

$x^3 - 1 \quad \text{(Matches with option H in the decoder)}$

Final Lock Combination:

• 1 → A
• 2 → N
• 3 → D
• 4 → H

Would you like more details on any step or have additional questions?

5 Related Questions:

1. How do you approach expanding binomials and trinomials efficiently?
2. What are common mistakes to avoid when multiplying polynomials?
3. How does the distributive property help in expanding polynomial expressions?
4. Can the perfect square method be applied to any binomial?
5. How does factoring relate to the multiplication of polynomials?

Tip:

When multiplying polynomials, always combine like terms carefully to avoid missing or adding terms incorrectly.