Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ 7. Which expression is the product of [tex]$(x+5)\left(2x^2-3x+5\right)$[/tex]?

A. [tex]2x^3 - 3x^2 + 5x + 25[/tex]

B. [tex]2x^3 + 7x^2 - 10x + 25[/tex]

C. [tex]2x^3 + 13x^2 - 20x + 25[/tex]

D. [tex]2x^3 + 10x^2 - 13x + 25[/tex]

To find the product of
[tex]$$
(x+5)(2x^2-3x+5),
$$[/tex]
we can use the distributive property (also called the FOIL method for binomials, extended here to a binomial times a trinomial).

Step 1. Multiply [tex]$x$[/tex] by each term in the second polynomial:

[tex]\[
\begin{aligned}
x \cdot 2x^2 &= 2x^3, \\
x \cdot (-3x) &= -3x^2, \\
x \cdot 5 &= 5x.
\end{aligned}
\][/tex]

Step 2. Multiply [tex]$5$[/tex] by each term in the second polynomial:

[tex]\[
\begin{aligned}
5 \cdot 2x^2 &= 10x^2, \\
5 \cdot (-3x) &= -15x, \\
5 \cdot 5 &= 25.
\end{aligned}
\][/tex]

Step 3. Combine the like terms:

Now, add all the products together:

[tex]\[
2x^3 + (-3x^2+10x^2) + (5x-15x) + 25.
\][/tex]

- The [tex]$x^2$[/tex] terms:
[tex]\[
-3x^2 + 10x^2 = 7x^2.
\][/tex]

- The [tex]$x$[/tex] terms:
[tex]\[
5x - 15x = -10x.
\][/tex]

Thus, the simplified result is:

[tex]\[
2x^3 + 7x^2 - 10x + 25.
\][/tex]

Final Answer: The product of [tex]$(x+5)(2x^2-3x+5)$[/tex] is

[tex]$$
2x^3 + 7x^2 - 10x + 25.
$$[/tex]