High School

Find the product of [tex]$2x^4(2x^2 + 3x + 4)$[/tex].

A. [tex]$2x^8 + 3x^4 + 4x^4$[/tex]

B. [tex][tex]$4x^6 + 6x^5 + 8x^4$[/tex][/tex]

C. [tex]$4x^4 + 3x^5 + 2x^6$[/tex]

D. [tex]$3x^6 + 4x^5 + 5x^4$[/tex]

Answer :

To find the product of
[tex]$$2x^4 \left(2x^2 + 3x + 4\right),$$[/tex]
we will use the distributive property by multiplying [tex]$2x^4$[/tex] with each term inside the parentheses.

1. Multiply [tex]$2x^4$[/tex] by [tex]$2x^2$[/tex]:
- Multiply the coefficients: [tex]$2 \times 2 = 4$[/tex]
- Add the exponents for [tex]$x$[/tex]: [tex]$4 + 2 = 6$[/tex]

This gives the term:
[tex]$$4x^6.$$[/tex]

2. Multiply [tex]$2x^4$[/tex] by [tex]$3x$[/tex]:
- Multiply the coefficients: [tex]$2 \times 3 = 6$[/tex]
- Add the exponents for [tex]$x$[/tex]: [tex]$4 + 1 = 5$[/tex]

This gives the term:
[tex]$$6x^5.$$[/tex]

3. Multiply [tex]$2x^4$[/tex] by [tex]$4$[/tex]:
- Multiply the coefficients: [tex]$2 \times 4 = 8$[/tex]
- The exponent remains [tex]$4$[/tex] since [tex]$4$[/tex] is a constant.

This gives the term:
[tex]$$8x^4.$$[/tex]

Finally, we combine all the terms to obtain the final product:
[tex]$$4x^6 + 6x^5 + 8x^4.$$[/tex]