Answer :
We want to multiply the two trinomials:
[tex]$$
(4x^2 - x - 4)(2x^2 + x + 12).
$$[/tex]
Step 1. Multiply each term in the first polynomial by every term in the second polynomial.
● Multiply the first term, [tex]$4x^2$[/tex], by each term in the second polynomial:
[tex]\[
4x^2 \cdot 2x^2 = 8x^4,
\][/tex]
[tex]\[
4x^2 \cdot x = 4x^3,
\][/tex]
[tex]\[
4x^2 \cdot 12 = 48x^2.
\][/tex]
● Multiply the second term, [tex]$-x$[/tex], by each term in the second polynomial:
[tex]\[
-x \cdot 2x^2 = -2x^3,
\][/tex]
[tex]\[
-x \cdot x = -x^2,
\][/tex]
[tex]\[
-x \cdot 12 = -12x.
\][/tex]
● Multiply the third term, [tex]$-4$[/tex], by each term in the second polynomial:
[tex]\[
-4 \cdot 2x^2 = -8x^2,
\][/tex]
[tex]\[
-4 \cdot x = -4x,
\][/tex]
[tex]\[
-4 \cdot 12 = -48.
\][/tex]
Step 2. Combine like terms.
● The [tex]$x^4$[/tex] term:
[tex]\[
8x^4.
\][/tex]
● The [tex]$x^3$[/tex] terms:
[tex]\[
4x^3 + (-2x^3)= 2x^3.
\][/tex]
● The [tex]$x^2$[/tex] terms:
[tex]\[
48x^2 + (-x^2) + (-8x^2)= 39x^2.
\][/tex]
● The [tex]$x$[/tex] terms:
[tex]\[
-12x + (-4x)= -16x.
\][/tex]
● The constant term:
[tex]\[
-48.
\][/tex]
Thus, the product is:
[tex]$$
8x^4 + 2x^3 + 39x^2 - 16x - 48.
$$[/tex]
Comparing with the given choices, the correct answer is option D.
[tex]$$
(4x^2 - x - 4)(2x^2 + x + 12).
$$[/tex]
Step 1. Multiply each term in the first polynomial by every term in the second polynomial.
● Multiply the first term, [tex]$4x^2$[/tex], by each term in the second polynomial:
[tex]\[
4x^2 \cdot 2x^2 = 8x^4,
\][/tex]
[tex]\[
4x^2 \cdot x = 4x^3,
\][/tex]
[tex]\[
4x^2 \cdot 12 = 48x^2.
\][/tex]
● Multiply the second term, [tex]$-x$[/tex], by each term in the second polynomial:
[tex]\[
-x \cdot 2x^2 = -2x^3,
\][/tex]
[tex]\[
-x \cdot x = -x^2,
\][/tex]
[tex]\[
-x \cdot 12 = -12x.
\][/tex]
● Multiply the third term, [tex]$-4$[/tex], by each term in the second polynomial:
[tex]\[
-4 \cdot 2x^2 = -8x^2,
\][/tex]
[tex]\[
-4 \cdot x = -4x,
\][/tex]
[tex]\[
-4 \cdot 12 = -48.
\][/tex]
Step 2. Combine like terms.
● The [tex]$x^4$[/tex] term:
[tex]\[
8x^4.
\][/tex]
● The [tex]$x^3$[/tex] terms:
[tex]\[
4x^3 + (-2x^3)= 2x^3.
\][/tex]
● The [tex]$x^2$[/tex] terms:
[tex]\[
48x^2 + (-x^2) + (-8x^2)= 39x^2.
\][/tex]
● The [tex]$x$[/tex] terms:
[tex]\[
-12x + (-4x)= -16x.
\][/tex]
● The constant term:
[tex]\[
-48.
\][/tex]
Thus, the product is:
[tex]$$
8x^4 + 2x^3 + 39x^2 - 16x - 48.
$$[/tex]
Comparing with the given choices, the correct answer is option D.
