Option A:
[tex]\left(x^{4}\right)\left(3 x^{3}-2\right)\left(4 x^{2}+5 x\right)=12 x^{9} +15 x^{8} - 8 x^{6}-10 x^{5}[/tex]
Solution:
Given expression [tex]\left(x^{4}\right)\left(3 x^{3}-2\right)\left(4 x^{2}+5 x\right)[/tex].
To find the product of the above expression:
[tex]\left(x^{4}\right)\left(3 x^{3}-2\right)\left(4 x^{2}+5 x\right)[/tex]
First multiply first two factors with each term.
[tex]=(x^{4} \times 3 x^{3}- x^{4} \times 2 ) \left(4 x^{2}+5 x\right)[/tex]
Using exponent rule: [tex]a^m \cdot a^n=a^{m+n}[/tex]
[tex]=(3 x^{7}- 2x^{4} ) \left(4 x^{2}+5 x\right)[/tex]
Now multiply these two factors with each term.
[tex]=3 x^{7} (4 x^{2}+5 x)- 2x^{4} \left(4 x^{2}+5 x\right)[/tex]
[tex]=(4 x^{2} \times 3 x^{7} +5 x \times 3 x^{7} )- \left(4 x^{2} \times 2x^{4}+5 x \times 2x^{4}\right)[/tex]
Using exponent rule: [tex]a^m \cdot a^n=a^{m+n}[/tex]
[tex]=(12 x^{9} +15 x^{8} )- (8 x^{6}+10 x^{5})[/tex]
[tex]=12 x^{9} +15 x^{8} - 8 x^{6}-10 x^{5}[/tex]
[tex]\left(x^{4}\right)\left(3 x^{3}-2\right)\left(4 x^{2}+5 x\right)=12 x^{9} +15 x^{8} - 8 x^{6}-10 x^{5}[/tex]
Hence option A is the correct answer.
