Question

1. choose the box method that has been completed correctly to multiply $(x - 3)(x^{2} + 3x + 9).$

Explanation:

Step1: Define box multiplication pairs

The box method multiplies each term of $(x-3)$ with each term of $(x^2+3x+9)$. The pairs are:

1. $x \times x^2 = x^3$
2. $x \times 3x = 3x^2$
3. $x \times 9 = 9x$
4. $-3 \times x^2 = -3x^2$
5. $-3 \times 3x = -9x$
6. $-3 \times 9 = -27$

Step2: Match pairs to box structure

The box has rows for $x, -3$ and columns for $x^2, 3x, 9$. Populate each cell:

• Row $x$, Column $x^2$: $x^3$
• Row $x$, Column $3x$: $3x^2$
• Row $x$, Column $9$: $9x$
• Row $-3$, Column $x^2$: $-3x^2$
• Row $-3$, Column $3x$: $-9x$
• Row $-3$, Column $9$: $-27$

Step3: Verify against options

Compare the populated cells to the given box options.

Answer:

The correct box is the left-most option, with cells:
Row $x$: $x^3$, $3x^2$, $9x$
Row $-3$: $-3x^2$, $-9x$, $-27$
(Top row labels: $x, -3$; Column labels: $x^2, 3x, 9$)