Question

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

Explanation:

First, let's rearrange the polynomial in descending order of exponents: $-3x^4 -4x^3 +6x^2 -3x -2$

Step1: Create box with terms

We make a 2-row, 5-column table. Row 1: $-6$, $-5x$; Column 1: $-3x^4$, $-4x^3$, $6x^2$, $-3x$, $-2$

Step2: Calculate each box product

1. $-6 \times (-3x^4) = 18x^4$
2. $-6 \times (-4x^3) = 24x^3$
3. $-6 \times 6x^2 = -36x^2$
4. $-6 \times (-3x) = 18x$
5. $-6 \times (-2) = 12$
6. $-5x \times (-3x^4) = 15x^5$
7. $-5x \times (-4x^3) = 20x^4$
8. $-5x \times 6x^2 = -30x^3$
9. $-5x \times (-3x) = 15x^2$
10. $-5x \times (-2) = 10x$

Step3: Combine like terms

• $x^5$ term: $15x^5$
• $x^4$ terms: $18x^4 + 20x^4 = 38x^4$
• $x^3$ terms: $24x^3 - 30x^3 = -6x^3$
• $x^2$ terms: $-36x^2 + 15x^2 = -21x^2$
• $x$ terms: $18x + 10x = 28x$
• Constant term: $12$

Filled box (row × column):

	$-3x^4$	$-4x^3$	$6x^2$	$-3x$	$-2$
$-5x$	$15x^5$	$20x^4$	$-30x^3$	$15x^2$	$10x$

Answer:

Filled table:

	$-3x^4$	$-4x^3$	$6x^2$	$-3x$	$-2$
$-5x$	$15x^5$	$20x^4$	$-30x^3$	$15x^2$	$10x$

Simplified polynomial: $15x^5 + 38x^4 -6x^3 -21x^2 +28x +12$