Question

use the box method to distribute and simplify $(-4 + x^3 - x - 5x^2)(-6 - 4x)$. drag and drop the terms to the correct locations of the table. $(-4 + x^3 - x - 5x^2)(-6 - 4x)$ rows: columns:

Explanation:

Step1: Set up box dimensions

First polynomial has 4 terms, second has 2 terms. So we use a 4-row, 2-column table.

Step2: List terms for rows/columns

Row terms (from $-4 + x^3 - x - 5x^2$, ordered by degree): $x^3$, $-5x^2$, $-x$, $-4$
Column terms (from $-6 - 4x$): $-6$, $-4x$

Step3: Fill box with products

	$-6$	$-4x$
$-5x^2$	$-5x^2 \times (-6) = 30x^2$	$-5x^2 \times (-4x) = 20x^3$
$-x$	$-x \times (-6) = 6x$	$-x \times (-4x) = 4x^2$
$-4$	$-4 \times (-6) = 24$	$-4 \times (-4x) = 16x$

Step4: Combine like terms

1. $x^4$ term: $-4x^4$
2. $x^3$ terms: $-6x^3 + 20x^3 = 14x^3$
3. $x^2$ terms: $30x^2 + 4x^2 = 34x^2$
4. $x$ terms: $6x + 16x = 22x$
5. Constant term: $24$

Step5: Write simplified polynomial

Arrange terms in descending degree order.

Answer:

Filled Box Table:

	$-6$	$-4x$
$-5x^2$	$30x^2$	$20x^3$
$-x$	$6x$	$4x^2$
$-4$	$24$	$16x$

Simplified Polynomial:

$-4x^4 + 14x^3 + 34x^2 + 22x + 24$

Table Dimensions:

Rows: 4, Columns: 2