Answer :
To multiply the polynomials
[tex]$$\left(x^2-5x\right)\left(2x^2+x-3\right),$$[/tex]
we start by using the distributive property.
1. Multiply each term in the first polynomial by each term in the second polynomial.
Multiply [tex]$x^2$[/tex] by each term in [tex]$(2x^2 + x -3)$[/tex]:
[tex]$$x^2 \cdot 2x^2 = 2x^4,$$[/tex]
[tex]$$x^2 \cdot x = x^3,$$[/tex]
[tex]$$x^2 \cdot (-3) = -3x^2.$$[/tex]
So, the first group of terms is:
[tex]$$2x^4 + x^3 - 3x^2.$$[/tex]
2. Next, multiply [tex]$-5x$[/tex] by each term in [tex]$(2x^2 + x -3)$[/tex]:
[tex]$$-5x \cdot 2x^2 = -10x^3,$$[/tex]
[tex]$$-5x \cdot x = -5x^2,$$[/tex]
[tex]$$-5x \cdot (-3) = 15x.$$[/tex]
So, the second group of terms is:
[tex]$$-10x^3 - 5x^2 + 15x.$$[/tex]
3. Now, add the two results together:
[tex]$$\begin{align*}
(2x^4 + x^3 - 3x^2) + (-10x^3 - 5x^2 + 15x) &= 2x^4 + (x^3 - 10x^3) + (-3x^2 - 5x^2) + 15x \\
&= 2x^4 - 9x^3 - 8x^2 + 15x.
\end{align*}$$[/tex]
Thus, the final expanded expression is
[tex]$$2x^4 - 9x^3 - 8x^2 + 15x.$$[/tex]
The correct answer is Option C.
[tex]$$\left(x^2-5x\right)\left(2x^2+x-3\right),$$[/tex]
we start by using the distributive property.
1. Multiply each term in the first polynomial by each term in the second polynomial.
Multiply [tex]$x^2$[/tex] by each term in [tex]$(2x^2 + x -3)$[/tex]:
[tex]$$x^2 \cdot 2x^2 = 2x^4,$$[/tex]
[tex]$$x^2 \cdot x = x^3,$$[/tex]
[tex]$$x^2 \cdot (-3) = -3x^2.$$[/tex]
So, the first group of terms is:
[tex]$$2x^4 + x^3 - 3x^2.$$[/tex]
2. Next, multiply [tex]$-5x$[/tex] by each term in [tex]$(2x^2 + x -3)$[/tex]:
[tex]$$-5x \cdot 2x^2 = -10x^3,$$[/tex]
[tex]$$-5x \cdot x = -5x^2,$$[/tex]
[tex]$$-5x \cdot (-3) = 15x.$$[/tex]
So, the second group of terms is:
[tex]$$-10x^3 - 5x^2 + 15x.$$[/tex]
3. Now, add the two results together:
[tex]$$\begin{align*}
(2x^4 + x^3 - 3x^2) + (-10x^3 - 5x^2 + 15x) &= 2x^4 + (x^3 - 10x^3) + (-3x^2 - 5x^2) + 15x \\
&= 2x^4 - 9x^3 - 8x^2 + 15x.
\end{align*}$$[/tex]
Thus, the final expanded expression is
[tex]$$2x^4 - 9x^3 - 8x^2 + 15x.$$[/tex]
The correct answer is Option C.
