To solve the problem of multiplying [tex]\( 5 x^2 \)[/tex] by the polynomial [tex]\( 2 x^2 + 13 x - 5 \)[/tex], let's proceed step-by-step:
1. Identify the given polynomial and the term that multiplies it:
- The polynomial is [tex]\( 2 x^2 + 13 x - 5 \)[/tex].
- The term that multiplies the polynomial is [tex]\( 5 x^2 \)[/tex].
2. Distribute [tex]\( 5 x^2 \)[/tex] to each term in the polynomial:
This involves multiplying [tex]\( 5 x^2 \)[/tex] by each term of the polynomial separately and then combining the results.
Let's do the multiplication term-by-term:
- Multiply [tex]\( 5 x^2 \)[/tex] by [tex]\( 2 x^2 \)[/tex]:
[tex]\[
5 x^2 \cdot 2 x^2 = 10 x^4
\][/tex]
- Multiply [tex]\( 5 x^2 \)[/tex] by [tex]\( 13 x \)[/tex]:
[tex]\[
5 x^2 \cdot 13 x = 65 x^3
\][/tex]
- Multiply [tex]\( 5 x^2 \)[/tex] by [tex]\( -5 \)[/tex]:
[tex]\[
5 x^2 \cdot (-5) = -25 x^2
\][/tex]
3. Combine the resulting terms to form the final polynomial:
When you put all these terms together, you get:
[tex]\[
10 x^4 + 65 x^3 - 25 x^2
\][/tex]
Thus, the resulting polynomial after multiplying [tex]\( 5 x^2 \)[/tex] by [tex]\( 2 x^2 + 13 x - 5 \)[/tex] is:
[tex]\[
10 x^4 + 65 x^3 - 25 x^2
\][/tex]
Therefore, the correct choice among the given options is:
[tex]\[
\boxed{10 x^4 + 65 x^3 - 25 x^2}
\][/tex]
