Let's simplify the given expression step-by-step:
Given:
[tex]\[ 8a^2 + 3ab - 2b^2 \cdot 3a^2 - ab + 5b^2 + 5ab - 7a^2 - b^2 \][/tex]
First, combine the like terms.
Step 1: Simplify each part of the expression
Combine the quadratic terms involving [tex]\( a^2 \)[/tex]:
[tex]\[ 8a^2 - 6a^2b^2 - 7a^2 \][/tex]
Combine the linear terms involving [tex]\( ab \)[/tex]:
[tex]\[ 3ab - ab + 5ab \][/tex]
Combine the quadratic terms involving [tex]\( b^2 \)[/tex]:
[tex]\[ -2b^2 \cdot 3a^2 + 5b^2 - b^2 \][/tex]
Step 2: Combine each group
1. For the [tex]\( a^2 \)[/tex] terms:
[tex]\[ 8a^2 - 7a^2 - 6a^2b^2 = a^2 - 6a^2b^2 \][/tex]
2. For the [tex]\( ab \)[/tex] terms:
[tex]\[ 3ab - ab + 5ab = 7ab \][/tex]
3. For the [tex]\( b^2 \)[/tex] terms:
[tex]\[ 5b^2 - b^2 = 4b^2 \][/tex]
Step 3: Combine the simplified terms
Putting it all together, we get:
[tex]\[ -6a^2b^2 + a^2 + 7ab + 4b^2 \][/tex]
So, the simplified form of the expression [tex]\( 8a^2 + 3ab - 2b^2 \cdot 3a^2 - ab + 5b^2 + 5ab - 7a^2 - b^2 \)[/tex] is:
[tex]\[ -6a^2b^2 + a^2 + 7ab + 4b^2 \][/tex]
That’s the final simplified expression.
