Question

question
use the box method to distribute and simplify ((3x - 1)(4x^4 - 4x - 5x^3 - 5 + 2x^2)). drag and drop the terms to the correct locations of the table.
((3x - 1)(4x^4 - 4x - 5x^3 - 5 + 2x^2))
rows:
columns:

Explanation:

Step1: Rearrange polynomial by degree

First, order the second polynomial from highest to lowest degree:
$4x^4 - 5x^3 + 2x^2 - 4x - 5$

Step2: Set up box table

Create a table with 2 rows (for $3x$ and $-1$) and 5 columns (for each term in the ordered polynomial):

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

Step3: Multiply row 1 terms

Calculate products for the $3x$ row:

• $3x \times 4x^4 = 12x^5$
• $3x \times (-5x^3) = -15x^4$
• $3x \times 2x^2 = 6x^3$
• $3x \times (-4x) = -12x^2$
• $3x \times (-5) = -15x$

Step4: Multiply row 2 terms

Calculate products for the $-1$ row:

• $-1 \times 4x^4 = -4x^4$
• $-1 \times (-5x^3) = 5x^3$
• $-1 \times 2x^2 = -2x^2$
• $-1 \times (-4x) = 4x$
• $-1 \times (-5) = 5$

Step5: Fill the box table

Populate the table with the calculated products:

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

Step6: Combine like terms

Sum all terms and combine like terms:
$12x^5 + (-15x^4 -4x^4) + (6x^3 +5x^3) + (-12x^2 -2x^2) + (-15x +4x) +5$
$=12x^5 -19x^4 +11x^3 -14x^2 -11x +5$

Answer:

Completed Box Table:

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

Simplified Polynomial:

$12x^5 -19x^4 +11x^3 -14x^2 -11x +5$