2. Find the products. \( \begin{array}{ll}\text { a) }(x+1)(x+3) & \text { b) }(a+3)(a+4) \\ \text { c) }\left(x^{2}-1\right)\left(2 x^{2}+3\right) & \text { d) }\left(a^{2}-2\right)\left(a^{2}+2\right) \\ \text { e) }(3 x y+2 z)(2 x y-3 z) & \text { f) }(p r+2 t)(3 p r-t) \\ \text { g) }\left(x^{2}-4\right)\left(4 x^{2}-3\right) & \text { h) }\left(2 a^{2}+1\right)\left(3 a^{2}+1\right) \\ \text { i) }\left(x+\frac{1}{2}\right)\left(x+\frac{1}{4}\right) & \text { j) }\left(2 x+\frac{1}{3}\right)\left(3 x-\frac{1}{2}\right) \\ \text { k) }(2 a-b)(2 a+b) & \text { l) }(3 x+2 y)(3 x-2 y) \\ \text { m) }\left(\frac{1}{2} x+\frac{1}{3} y\right)\left(\frac{1}{3} x-\frac{1}{2} y\right) & \text {-n) }(x+3)^{2}\end{array} \)

To find the product of each expression, we can apply the distributive property, also known as the FOIL method for binomials, and use identities for special products where applicable. a) \((x + 1)(x + 3) = x^2 + 4x + 3\) b) \((a + 3)(a + 4) = a^2 + 7a + 12\) c) \((x^2 - 1)(2x^2 + 3) = 2x^4 + 3x^2 - 2x^2 - 3 = 2x^4 + x^2 - 3\) d) \((a^2 - 2)(a^2 + 2) = a^4 - 4\) (This is a difference of squares.) e) \((3xy + 2z)(2xy - 3z) = 6x^2y^2 - 9xyz + 4xz - 6z^2\) f) \((pr + 2t)(3pr - t) = 3p^2r^2 + 6ptr - pr - 2t^2 = 3p^2r^2 + (6t - p)r\) g) \((x^2 - 4)(4x^2 - 3) = 4x^4 - 16x^2 - 3x^2 + 12 = 4x^4 - 19x^2 + 12\) h) \((2a^2 + 1)(3a^2 + 1) = 6a^4 + 2a^2 + 3a^2 + 1 = 6a^4 + 5a^2 + 1\) i) \(\left(x + \frac{1}{2}\right)\left(x + \frac{1}{4}\right) = x^2 + \frac{3}{4}x + \frac{1}{8}\) j) \(\left(2x + \frac{1}{3}\right)\left(3x - \frac{1}{2}\right) = 6x^2 - \frac{4}{3}x + \frac{3}{6} = 6x^2 - \frac{4}{3}x + \frac{1}{2}\) k) \((2a - b)(2a + b) = 4a^2 - b^2\) (This is a difference of squares.) l) \((3x + 2y)(3x - 2y) = 9x^2 - 4y^2\) (This is also a difference of squares.) m) \(\left(\frac{1}{2}x + \frac{1}{3}y\right)\left(\frac{1}{3}x - \frac{1}{2}y\right) = \frac{1}{6}x^2 - \frac{1}{4}xy + \frac{1}{6}xy - \frac{1}{6}y^2 = \frac{1}{6}x^2 - \frac{1}{12}xy - \frac{1}{6}y^2\) -n) \((x + 3)^2 = x^2 + 6x + 9\) (This is a perfect square.)