Simplify [tex]2 a^2 b^3\left(4 a^2 + 3 a b^2 - a b\right)[/tex]:

A. [tex]8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4[/tex]

B. [tex]8 a^4 b^5 + 3 a^3 b^5 + 2 a^3 b^4[/tex]

C. [tex]8 a^4 b^5 + 3 a^3 b^5 - 2 a^3 b^4[/tex]

D. [tex]8 a^4 b^3 + 6 a^3 b^5 + 2 a^3 b^4[/tex]



Answer :

Sure, let's go through the detailed steps to simplify the given expression.

We need to simplify [tex]\( 2 a^2 b^3 \left( 4 a^2 + 3 a b^2 - a b \right) \)[/tex].

Step 1: Distribute [tex]\(2 a^2 b^3\)[/tex] to each term inside the parentheses:

[tex]\[ 2 a^2 b^3 \cdot 4 a^2 \][/tex]

[tex]\[ 2 a^2 b^3 \cdot 3 a b^2 \][/tex]

[tex]\[ 2 a^2 b^3 \cdot (- a b) \][/tex]

Step 2: Simplify each term:

- For the first term, multiply:

[tex]\[ 2 a^2 b^3 \cdot 4 a^2 = 2 \cdot 4 \cdot a^2 \cdot a^2 \cdot b^3 = 8 a^4 b^3 \][/tex]

- For the second term, multiply:

[tex]\[ 2 a^2 b^3 \cdot 3 a b^2 = 2 \cdot 3 \cdot a^2 \cdot a \cdot b^3 \cdot b^2 = 6 a^3 b^5 \][/tex]

- For the third term, multiply:

[tex]\[ 2 a^2 b^3 \cdot (- a b) = 2 \cdot (-1) \cdot a^2 \cdot a \cdot b^3 \cdot b = -2 a^3 b^4 \][/tex]

Step 3: Combine all simplified terms:

[tex]\[ 8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4 \][/tex]

Conclusion:

Therefore, the simplified expression is [tex]\( 8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4 \)[/tex].

Looking at the given choices, the correct answer is:

A. [tex]\( 8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4 \)[/tex]