(n+2)(n^2+5n-3) (3n-1)(2n^2+4n+4) (2n+3)(6n^2-2n+1) (4n-5)(n^2-7n-2) (3n-4)(4n^2+2n+3 (n+8)(6n^2-n-4)

expand each of the given expressions step by step.<br /><br />1. \((n+2)(n^2 + 5n - 3)\)<br /><br />Using the distributive property (FOIL method):<br />\[<br />(n+2)(n^2 + 5n - 3) = n(n^2 + 5n - 3) + 2(n^2 + 5n - 3)<br />\]<br />\[<br />= n^3 + 5n^2 - 3n + 2n^2 + 10n - 6<br />\]<br />\[<br />= n^3 + 7n^2 + 7n - 6<br />\]<br /><br />2. \((3n-1)(2n^2 + 4n + 4)\)<br /><br />Using the distributive property:<br />\[<br />(3n-1)(2n^2 + 4n + 4) = 3n(2n^2 + 4n + 4) - 1(2n^2 + 4n + 4)<br />\]<br />\[<br />= 6n^3 + 12n^2 + 12n - 2n^2 - 4n - 4<br />\]<br />\[<br />= 6n^3 + 10n^2 + 8n - 4<br />\]<br /><br />3. \((2n+3)(6n^2 - 2n + 1)\)<br /><br />Using the distributive property:<br />\[<br />(2n+3)(6n^2 - 2n + 1) = 2n(6n^2 - 2n + 1) + 3(6n^2 - 2n + 1)<br />\]<br />\[<br />= 12n^3 - 4n^2 + 2n + 18n^2 - 6n + 3<br />\]<br />\[<br />= 12n^3 + 14n^2 - 4n + 3<br />\]<br /><br />4. \((4n-5)(n^2 - 7n - 2)\)<br /><br />Using the distributive property:<br />\[<br />(4n-5)(n^2 - 7n - 2) = 4n(n^2 - 7n - 2) - 5(n^2 - 7n - 2)<br />\]<br />\[<br />= 4n^3 - 28n^2 - 8n - 5n^2 + 35n + 10<br />\]<br />\[<br />= 4n^3 - 33n^2 + 27n + 10<br />\]<br /><br />5. \((3n-4)(4n^2 + 2n + 3)\)<br /><br />Using the distributive property:<br />\[<br />(3n-4)(4n^2 + 2n + 3) = 3n(4n^2 + 2n + 3) - 4(4n^2 + 2n + 3)<br />\]<br />\[<br />= 12n^3 + 6n^2 + 9n - 16n^2 - 8n - 12<br />\]<br />\[<br />= 12n^3 - 10n^2 + 1n - 12<br />\]<br /><br />6. \((n+8)(6n^2 - n - 4)\)<br /><br />Using the distributive property:<br />\[<br />(n+8)(6n^2 - n - 4) = n(6n^2 - n - 4) + 8(6n^2 - n - 4)<br />\]<br />\[<br />= 6n^3 - n^2 - 4n + 48n^2 - 8n - 32<br />\]<br />\[<br />= 6n^3 + 47n^2 - 12n - 32<br />\]<br /><br />So, the expanded forms of the given expressions are:<br /><br />1. \(n^3 + 7n^2 + 7n - 6\)<br />2. \(6n^3 + 10n^2 + 8n - 4\)<br />3. \(12n^3 + 14n^2 - 4n + 3\)<br />4. \(4n^3 - 33n^2 + 27n + 10\)<br />5. \(12n^3 - 10n^2 + 1n - 12\)<br />6. \(6n^3 + 47n^2 - 12n - 32\)