(4a-7)(3a-2) (2a+5)(2a-5) (6a-1)(2a+4) (a+2b)(4a+b) (5a+3b)(a-4b) I (3a-8b)(2a-b)

Let's expand each of the given expressions step by step.<br /><br />1. \((4a-7)(3a-2)\)<br /><br />Using the distributive property (FOIL method):<br />\[<br />(4a-7)(3a-2) = 4a \cdot 3a + 4a \cdot (-2) + (-7) \cdot 3a + (-7) \cdot (-2)<br />\]<br />\[<br />= 12a^2 - 8a - 21a + 14<br />\]<br />\[<br />= 12a^2 - 29a + 14<br />\]<br /><br />2. \((2a+5)(2a-5)\)<br /><br />Using the difference of squares formula:<br />\[<br />(2a+5)(2a-5) = (2a)^2 - (5)^2<br />\]<br />\[<br />= 4a^2 - 25<br />\]<br /><br />3. \((6a-1)(2a+4)\)<br /><br />Using the distributive property:<br />\[<br />(6a-1)(2a+4) = 6a \cdot 2a + 6a \cdot 4 + (-1) \cdot 2a + (-1) \cdot 4<br />\]<br />\[<br />= 12a^2 + 24a - 2a - 4<br />\]<br />\[<br />= 12a^2 + 22a - 4<br />\]<br /><br />4. \((a+2b)(4a+b)\)<br /><br />Using the distributive property:<br />\[<br />(a+2b)(4a+b) = a \cdot 4a + a \cdot b + 2b \cdot 4a + 2b \cdot b<br />\]<br />\[<br />= 4a^2 + ab + 8ab + 2b^2<br />\]<br />\[<br />= 4a^2 + 9ab + 2b^2<br />\]<br /><br />5. \((5a+3b)(a-4b)\)<br /><br />Using the distributive property:<br />\[<br />(5a+3b)(a-4b) = 5a \cdot a + 5a \cdot (-4b) + 3b \cdot a + 3b \cdot (-4b)<br />\]<br />\[<br />= 5a^2 - 20ab + 3ab - 12b^2<br />\]<br />\[<br />= 5a^2 - 17ab - 12b^2<br />\]<br /><br />6. \((3a-8b)(2a-b)\)<br /><br />Using the distributive property:<br />\[<br />(3a-8b)(2a-b) = 3a \cdot 2a + 3a \cdot (-b) + (-8b) \cdot 2a + (-8b) \cdot (-b)<br />\]<br />\[<br />= 6a^2 - 3ab - 16ab + 8b^2<br />\]<br />\[<br />= 6a^2 - 19ab + 8b^2<br />\]<br /><br />So, the expanded forms of the given expressions are:<br /><br />1. \(12a^2 - 29a + 14\)<br />2. \(4a^2 - 25\)<br />3. \(12a^2 + 22a - 4\)<br />4. \(4a^2 + 9ab + 2b^2\)<br />5. \(5a^2 - 17ab - 12b^2\)<br />6. \(6a^2 - 19ab + 8b^2\)