Certainly! Let's solve the given expression [tex]\(\left(3 a^2 b + 4 a b^2\right)\left(a^2 - a b + b^2\right)\)[/tex] step by step.
1. Expression Expansion:
We'll start by distributing each term in the first expression across each term in the second expression. This involves applying the distributive property of multiplication over addition.
[tex]\[
\left(3 a^2 b + 4 a b^2\right)\left(a^2 - a b + b^2\right)
\][/tex]
We distribute [tex]\(3 a^2 b\)[/tex] across each term in the second expression:
[tex]\[
3 a^2 b \cdot a^2 + 3 a^2 b \cdot (-a b) + 3 a^2 b \cdot b^2
\][/tex]
And then distribute [tex]\(4 a b^2\)[/tex] across each term in the second expression:
[tex]\[
4 a b^2 \cdot a^2 + 4 a b^2 \cdot (-a b) + 4 a b^2 \cdot b^2
\][/tex]
2. Multiplying Each Term:
Let's multiply each pair of terms:
- [tex]\( 3 a^2 b \cdot a^2 = 3 a^4 b \)[/tex]
- [tex]\( 3 a^2 b \cdot (-a b) = -3 a^3 b^2 \)[/tex]
- [tex]\( 3 a^2 b \cdot b^2 = 3 a^2 b^3 \)[/tex]
- [tex]\( 4 a b^2 \cdot a^2 = 4 a^3 b^2 \)[/tex]
- [tex]\( 4 a b^2 \cdot (-a b) = -4 a^2 b^3 \)[/tex]
- [tex]\( 4 a b^2 \cdot b^2 = 4 a b^4 \)[/tex]
3. Combine Like Terms:
Now combine all these terms together:
[tex]\[
3 a^4 b + (-3 a^3 b^2) + 3 a^2 b^3 + 4 a^3 b^2 + (-4 a^2 b^3) + 4 a b^4
\][/tex]
4. Simplifying the Expression:
Grouping like terms, we get:
[tex]\[
3 a^4 b + (-3 a^3 b^2 + 4 a^3 b^2) + (3 a^2 b^3 - 4 a^2 b^3) + 4 a b^4
\][/tex]
Simplify inside the parentheses:
- For [tex]\( a^3 b^2 \)[/tex] terms: [tex]\( -3 a^3 b^2 + 4 a^3 b^2 = 1 a^3 b^2 \)[/tex]
- For [tex]\( a^2 b^3 \)[/tex] terms: [tex]\( 3 a^2 b^3 - 4 a^2 b^3 = -1 a^2 b^3 \)[/tex]
Putting it all together, we get:
[tex]\[
3 a^4 b + 1 a^3 b^2 - 1 a^2 b^3 + 4 a b^4
\][/tex]
5. Final Simplified Expression:
Thus, the simplified product of the given expressions is:
[tex]\[
3 a^4 b + a^3 b^2 - a^2 b^3 + 4 a b^4
\][/tex]
