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Advanced Math / Equivalent expressions Difficulty: Medium

Which expression is equivalent to ${(x^{2} + 11)}^{2} + (x - 5) (x + 5)$ ?

$x^{4} + 23 x^{2} - 14$

$x^{4} + 23 x^{2} + 96$

$x^{4} + 12 x^{2} + 121$

$x^{4} + x^{2} + 146$

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Explanation

Choice B is correct. The expression $left parenthesis x squared plus 11 right parenthesis squared$ can be written as $(x^{2} + 11) (x^{2} + 11)$ , which is equivalent to $x^{2} (x^{2} + 11) + 11 (x^{2} + 11)$ . Distributing $x^{2}$ and $11$ to $left parenthesis x squared plus 11 right parenthesis$ yields $x^{4} + 11 x^{2} + 11 x^{2} + 121$ , or $x^{4} + 22 x^{2} + 121$ . The expression $(x - 5) (x + 5)$ is equivalent to $(x - 5) x + (x - 5) 5$ . Distributing $x$ and $5$ to $left parenthesis x minus 5 right parenthesis$ yields $x^{2} - 5 x + 5 x - 25$ , or $x squared minus 25$ . Therefore, the expression ${(x^{2} + 11)}^{2} + (x - 5) (x + 5)$ is equivalent to $(x^{4} + 22 x^{2} + 121) + (x^{2} - 25)$ , or $x^{4} + 22 x^{2} + 121 + x^{2} - 25$ . Combining like terms in this expression yields $x^{4} + 23 x^{2} + 96$ .

Choice A is incorrect. Equivalent expressions must be equivalent for any value of $x$ . Substituting $0$ for $x$ in this expression yields $negative 14$ , whereas substituting $0$ for $x$ in the given expression yields $96$ .

Choice C is incorrect. Equivalent expressions must be equivalent for any value of $x$ . Substituting $0$ for $x$ in this expression yields $121$ , whereas substituting $0$ for $x$ in the given expression yields $96$ .

Choice D is incorrect. Equivalent expressions must be equivalent for any value of $x$ . Substituting $0$ for $x$ in this expression yields $146$ , whereas substituting $0$ for $x$ in the given expression yields $96$ .