Certainly! Let's simplify the given expression step-by-step:
Given expression:
[tex]\[ 3\left(5 b a^2 - 4\right)\left(5 b a^2 + 4\right) \][/tex]
### Step 1: Recognize the product of binomials
Notice that the expression inside the parentheses is in the form:
[tex]\[ \left(x - y\right)\left(x + y\right) \][/tex]
where [tex]\( x = 5ba^2 \)[/tex] and [tex]\( y = 4 \)[/tex].
### Step 2: Use the difference of squares formula
The product of binomials in the form [tex]\( (x - y)(x + y) \)[/tex] follows the difference of squares formula:
[tex]\[ (x - y)(x + y) = x^2 - y^2 \][/tex]
### Step 3: Apply the formula
Substituting [tex]\( x = 5ba^2 \)[/tex] and [tex]\( y = 4 \)[/tex] into the formula [tex]\( (x - y)(x + y) \)[/tex]:
[tex]\[ (5ba^2 - 4)(5ba^2 + 4) = (5ba^2)^2 - 4^2 \][/tex]
### Step 4: Simplify the expression
Calculate each part of the equation:
[tex]\[ (5ba^2)^2 = (5ba^2) \times (5ba^2) = 25b^2a^4 \][/tex]
[tex]\[ 4^2 = 16 \][/tex]
So, substituting back:
[tex]\[ (5ba^2 - 4)(5ba^2 + 4) = 25b^2a^4 - 16 \][/tex]
### Step 5: Multiply by 3
Now, we need to incorporate the multiplication by 3 from the original expression:
[tex]\[ 3 \left( 25b^2a^4 - 16 \right) \][/tex]
Distribute the 3:
[tex]\[ 3 \times 25b^2a^4 - 3 \times 16 \][/tex]
### Step 6: Final simplification
Perform the multiplications:
[tex]\[ 75b^2a^4 - 48 \][/tex]
### Conclusion:
Therefore, the simplified form of the given expression is:
[tex]\[ 75b^2a^4 - 48 \][/tex]
