Certainly! Let's simplify the expression step-by-step.
Given the expression:
[tex]\[ a^3 \left(2b - a^2 - 2b(a^3 - a^2) + 3a^3 - b(a^3 - b)\right) \][/tex]
we will simplify the inner expression inside the parentheses first.
### Step 1: Expand the inner terms
Let's start by expanding [tex]\(2b(a^3 - a^2)\)[/tex] and [tex]\(b(a^3 - b)\)[/tex]:
[tex]\[ 2b(a^3 - a^2) = 2ba^3 - 2ba^2 \][/tex]
[tex]\[ b(a^3 - b) = ba^3 - b^2 \][/tex]
Substituting these back into the expression:
[tex]\[ a^3 \left(2b - a^2 - (2ba^3 - 2ba^2) + 3a^3 - (ba^3 - b^2)\right) \][/tex]
### Step 2: Simplify each term
Substitute:
[tex]\[ 2b - a^2 - 2ba^3 + 2ba^2 + 3a^3 - ba^3 + b^2 \][/tex]
### Step 3: Combine like terms
Group and combine the like terms:
- Terms involving [tex]\(a^3\)[/tex]: [tex]\(3a^3 - 2ba^3 - ba^3 = 3a^3 - 3ba^3\)[/tex]
- Terms involving [tex]\(a^2\)[/tex]: [tex]\( - a^2 + 2ba^2 = (2b - 1)a^2\)[/tex]
- Constant terms: [tex]\(2b + b^2\)[/tex]
So, we can rewrite the expression inside the parentheses as:
[tex]\[ 3a^3 - 3ba^3 + (2b - 1)a^2 + 2b + b^2 \][/tex]
### Step 4: Factor the simplified expression
Now that the terms are consolidated, we multiply back by [tex]\(a^3\)[/tex]:
[tex]\[ a^3 \left(3a^3 - 3ba^3 + (2b-1)a^2 + 2b + b^2\right) = a^3 \left(3a^3 - a^2 - 2b(a^3 - a^2) + 2b + b^2\right) \][/tex]
Therefore, the simplified expression is:
[tex]\[ a^3 \left(3a^3 - 2a^2 b (a-1) - a^2 + 2b + b^2 \right) \][/tex]
In conclusion, we have:
Original expression:
[tex]\[ a^3\left(2 b - a^2 - 2 b\left(a^3 - a^2\right)+ 3 a^3 - b\left(a^3 - b\right) \right) \][/tex]
Simplified expression:
[tex]\[ a^3\left(3 a^3 - 2a^2 b (a-1) - a^2 - b(a^3 - b) + 2b\right) \][/tex]
Thus, the expression inside the parentheses has been simplified to:
[tex]\[ 3 a^3 - 2a^2 b (a - 1) - a^2 - b(a^3 - b) + 2b \][/tex] and the overall simplified form is:
[tex]\[ a^3 \left(3 a^3 - 2a^2 b (a - 1) - a^2 - b(a^3 - b) + 2b\right) \][/tex]
