d+14 (13) (n+2)(n^2+5n-3) (14) (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)

Let's expand each of the given expressions step by step.<br /><br />### (13) \((n+2)(n^2+5n-3)\)<br /><br />Using the distributive property (also known as the FOIL method for binomials):<br /><br />\[<br />(n+2)(n^2+5n-3) = n(n^2) + n(5n) + n(-3) + 2(n^2) + 2(5n) + 2(-3)<br />\]<br /><br />\[<br />= n^3 + 5n^2 - 3n + 2n^2 + 10n - 6<br />\]<br /><br />Combine like terms:<br /><br />\[<br />= n^3 + 7n^2 + 7n - 6<br />\]<br /><br />### (14) \((3n-1)(2n^2+4n+4)\)<br /><br />Again, using the distributive property:<br /><br />\[<br />(3n-1)(2n^2+4n+4) = 3n(2n^2) + 3n(4n) + 3n(4) - 1(2n^2) - 1(4n) - 1(4)<br />\]<br /><br />\[<br />= 6n^3 + 12n^2 + 12n - 2n^2 - 4n - 4<br />\]<br /><br />Combine like terms:<br /><br />\[<br />= 6n^3 + 10n^2 + 8n - 4<br />\]<br /><br />### \((2n+3)(6n^2-2n+1)\)<br /><br />Using the distributive property:<br /><br />\[<br />(2n+3)(6n^2-2n+1) = 2n(6n^2) + 2n(-2n) + 2n(1) + 3(6n^2) + 3(-2n) + 3(1)<br />\]<br /><br />\[<br />= 12n^3 - 4n^2 + 2n + 18n^2 - 6n + 3<br />\]<br /><br />Combine like terms:<br /><br />\[<br />= 12n^3 + 14n^2 - 4n + 3<br />\]<br /><br />### \((4n-5)(n^2-7n-2)\)<br /><br />Using the distributive property:<br /><br />\[<br />(4n-5)(n^2-7n-2) = 4n(n^2) + 4n(-7n) + 4n(-2) - 5(n^2) - 5(-7n) - 5(-2)<br />\]<br /><br />\[<br />= 4n^3 - 28n^2 - 8n - 5n^2 + 35n + 10<br />\]<br /><br />Combine like terms:<br /><br />\[<br />= 4n^3 - 33n^2 + 27n + 10<br />\]<br /><br />### \((3n-4)(4n^2+2n+3)\)<br /><br />Using the distributive property:<br /><br />\[<br />(3n-4)(4n^2+2n+3) = 3n(4n^2) + 3n(2n) + 3n(3) - 4(4n^2) - 4(2n) - 4(3)<br />\]<br /><br />\[<br />= 12n^3 + 6n^2 + 9n - 16n^2 - 8n - 12<br />\]<br /><br />Combine like terms:<br /><br />\[<br />= 12n^3 - 10n^2 + 1n - 12<br />\]<br /><br />### \((n+8)(6n^2-n-4)\)<br /><br />Using the distributive property:<br /><br />\[<br />(n+8)(6n^2-n-4) = n(6n^2) + n(-n) + n(-4) + 8(6n^2) + 8(-n) + 8(-4)<br />\]<br /><br />\[<br />= 6n^3 - n^2 - 4n + 48n^2 - 8n - 32<br />\]<br /><br />Combine like terms:<br /><br />\[<br />= 6n^3 + 47n^2 - 12n - 32<br />\]<br /><br />So, the expanded forms of the given expressions are:<br /><br />1. \((n+2)(n^2+5n-3) = n^3 + 7n^2 + 7n - 6\)<br />2. \((3n-1)(2n^2+4n+4) = 6n^3 + 10n^2 + 8n - 4\)<br />3. \((2n+3)(6n^2-2n+1) = 12n