Multiply [tex]\( 5x^2(2x^2 + 13x - 5) \)[/tex].

A. [tex]\( 10x^4 + 65x^3 - 25x^2 \)[/tex]
B. [tex]\( 10x^2 + 65x - 25 \)[/tex]
C. [tex]\( 7x^2 + 18x - 10 \)[/tex]
D. [tex]\( 7x^4 + 18x^3 - 10x^2 \)[/tex]



Answer :

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]