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Advanced Math / Equivalent expressions Difficulty: Medium

$open parenthesis, 2 x plus 5, close parenthesis, squared, minus, open parenthesis, x minus 2, close parenthesis, plus 2 times, open parenthesis, x plus 3, close parenthesis$

Which of the following is equivalent to the expression above?

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Explanation

Choice A is correct. The given expression can be rewritten as $open parenthesis, 2 x plus 5, close parenthesis, squared, plus, negative 1 times, open parenthesis, x minus 2, close parenthesis, plus, 2 times, open parenthesis, x plus 3, close parenthesis$ . Applying the distributive property, the expression $negative 1 times, open parenthesis, x minus 2, close parenthesis, plus, 2 times, open parenthesis, x plus 3, close parenthesis$ can be rewritten as $negative one times x, plus, negative 1 times negative 2, plus, 2 times x, plus, 2 times 3$ , or $negative x plus 2, plus 2 x, plus 6$ . Adding like terms yields $x plus 8$ . Substituting $x plus 8$ for $negative 1, times, open parenthesis, x minus 2, close parenthesis, plus, 2 times, open parenthesis, x plus 3, close parenthesis$ in the given expression yields $open parenthesis, 2 x plus 5, close parenthesis, squared, plus x, plus 8$ . By the rules of exponents, the expression $open parenthesis, 2 x plus 5, close parenthesis, squared$ is equivalent to $open parenthesis, 2 x plus 5, close parenthesis, times, open parenthesis, 2 x plus 5, close parenthesis$ . Applying the distributive property, this expression can be rewritten as $2 x times 2x, plus, 2 x times 5, plus, 5 times 2 x, plus, 5 times 5$ , or $4 x squared, plus 10 x, plus 10 x, plus 25$ . Adding like terms gives $4 x squared, plus 20 x, plus 25$ . Substituting $4 x squared, plus 20 x, plus 25$ for $open parenthesis, 2 x plus 5, close parenthesis, squared$ in the rewritten expression yields $4 x squared, plus 20 x, plus 25, plus x, plus 8$ , and adding like terms yields $4 x squared, plus 21 x, plus 33$ .

Choices B, C, and D are incorrect. Choices C and D may result from rewriting the expression $open parenthesis, 2 x plus 5, close parenthesis, squared$ as $4 x squared, plus 25$ , instead of as $4 x squared, plus 20 x, plus 25$ . Choices B and C may result from rewriting the expression $negative, open parenthesis, x minus 2, close parenthesis$ as $negative x minus 2$ , instead of $negative x plus 2$ .