EXAMPLE 1 Expand and simplify the following: \( \begin{array}{ll}\text { (a) }(3 x+2 y)(3 x-2 y) & \text { (b) }(a-5 b)(a+5 b) \\ \text { (c) }\left(4 x^{4}-3 y^{3}\right)\left(4 x^{4}+3 y^{3}\right) \\ \text { (d) } \quad(a-b+d)(a-b-d) & \end{array} \)

Sure! Let’s expand and simplify each of the expressions step by step. **(a)** \((3x + 2y)(3x - 2y)\) Using the difference of squares formula, we can expand it: \[ (3x)^2 - (2y)^2 = 9x^2 - 4y^2 \] **(b)** \((a - 5b)(a + 5b)\) Again applying the difference of squares: \[ a^2 - (5b)^2 = a^2 - 25b^2 \] **(c)** \((4x^{4} - 3y^{3})(4x^{4} + 3y^{3})\) Using the difference of squares once more: \[ (4x^{4})^2 - (3y^{3})^2 = 16x^{8} - 9y^{6} \] **(d)** \((a - b + d)(a - b - d)\) We can recognize this as a difference of squares on \( (a - b) \): \[ (a - b)^2 - d^2 = (a - b)^2 - d^2 \] Now expanding \((a - b)^2\): \[ (a - b)^2 = a^2 - 2ab + b^2 \] So the final result is: \[ a^2 - 2ab + b^2 - d^2 \] And voila! We’ve expanded and simplified all the given expressions!