Final answer:
To simplify the expression (4a^2 - 3b^2)(16a^4 + 12a^2b^2 + 9b^4), recognize it as the difference of squares: (4a^2 - 3b^2) × (4a^2 + 3b^2), resulting in 16a^4 - 9b^4.
Step-by-step explanation:
The student asked to simplify the expression (4a^2 - 3b^2)(16a^4 + 12a^2b^2 + 9b^4). This type of problem involves multiplying two polynomials together. To simplify this, we can recognize that the second polynomial is a perfect square trinomial, being the square of (4a^2 + 3b^2). We leverage this and recognize the entire expression as the difference of squares:
- (4a^2 - 3b^2) × (4a^2 + 3b^2)
- (4a^2)^2 - (3b^2)^2
- 16a^4 - 9b^4
So the simplified form of the given expression is 16a^4 - 9b^4.
When faced with polynomials, it's often helpful to look for patterns such as the difference of squares or perfect square trinomials to simplify expressions. Additionally, always remember to check your answers to ensure they are reasonable.