Option D:
[tex]\left(y^{2}+3 y+7\right)\left(8 y^{2}+y+1\right)=8 y^{4}+25 y^{3}+60 y^{2}+10y+7[/tex]
Solution:
Given expression is [tex]\left(y^{2}+3 y+7\right)\left(8 y^{2}+y+1\right)[/tex].
To find the product of the expression:
[tex]\left(y^{2}+3 y+7\right)\left(8 y^{2}+y+1\right)[/tex]
Multiply each term of the first term with each term of the 2nd term.
[tex]=y^{2}\left(8 y^{2}+y+1\right) +3 y\left(8 y^{2}+y+1\right) +7\left(8 y^{2}+y+1\right)[/tex]
Using the exponent rule: [tex]a^m \cdot a^n = a^{m+n}[/tex]
[tex]=\left(8 y^{4}+y^3+y^2\right) +\left(24 y^{3}+3y^2+3y\right) +\left(56 y^{2}+7y+7\right)[/tex]
[tex]=8 y^{4}+y^3+y^2+24 y^{3}+3y^2+3y+56 y^{2}+7y+7[/tex]
Arrange the terms with same power.
[tex]=8 y^{4}+y^3+24 y^{3}+y^2+3y^2+56 y^{2}+7y+3y+7[/tex]
[tex]=8 y^{4}+25 y^{3}+60 y^{2}+10y+7[/tex]
Hence option D is the correct answer.
[tex]\left(y^{2}+3 y+7\right)\left(8 y^{2}+y+1\right)=8 y^{4}+25 y^{3}+60 y^{2}+10y+7[/tex]
