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Easy

Question

Simplify the following expression.
open parentheses fraction numerator 2 x minus 1 over denominator x plus 3 end fraction minus fraction numerator x squared minus 4 over denominator 2 x plus 1 end fraction close parentheses space cross times space fraction numerator 2 x squared plus 7 x plus 3 over denominator x plus 2 end fraction

hintHint:

The expansions of certain identities are:
open parentheses x squared minus a squared close parentheses equals left parenthesis x plus a right parenthesis left parenthesis x minus a right parenthesis
left parenthesis x plus a right parenthesis left parenthesis x plus b right parenthesis equals x squared plus left parenthesis a plus b right parenthesis x plus a b
We are asked to simplify the given expression

The correct answer is: the simplified expression is (x2 -x3 +4x +11)/(x+2)


    Step 1 of 2:
    Fins the LCM of the expression open parentheses fraction numerator 2 x minus 1 over denominator x plus 3 end fraction minus fraction numerator x squared minus 4 over denominator 2 x plus 1 end fraction close parentheses,
    open parentheses fraction numerator 2 x minus 1 over denominator x plus 3 end fraction minus fraction numerator x squared minus 4 over denominator 2 x plus 1 end fraction close parentheses equals fraction numerator left parenthesis 2 x minus 1 right parenthesis left parenthesis 2 x plus 1 right parenthesis minus open parentheses x squared minus 4 close parentheses left parenthesis x plus 3 right parenthesis over denominator left parenthesis x plus 3 right parenthesis left parenthesis 2 x plus 1 right parenthesis end fraction
    equals fraction numerator left parenthesis 2 x right parenthesis squared minus 1 squared minus open parentheses x squared minus 4 close parentheses left parenthesis x plus 3 right parenthesis over denominator left parenthesis x plus 3 right parenthesis left parenthesis 2 x plus 1 right parenthesis end fraction
    equals fraction numerator open parentheses 4 x squared minus 1 close parentheses minus open parentheses x cubed plus 3 x squared minus 4 x minus 12 close parentheses over denominator left parenthesis x plus 3 right parenthesis left parenthesis 2 x plus 1 right parenthesis end fraction
    equals fraction numerator 4 x squared minus 1 minus x cubed minus 3 x squared plus 4 x plus 12 over denominator left parenthesis x plus 3 right parenthesis left parenthesis 2 x plus 1 right parenthesis end fraction
    equals fraction numerator x squared minus x cubed plus 4 x plus 11 over denominator left parenthesis x plus 3 right parenthesis left parenthesis 2 x plus 1 right parenthesis end fraction
    Thus, the expression becomes fraction numerator x squared minus x cubed plus 4 x plus 11 over denominator left parenthesis x plus 3 right parenthesis left parenthesis 2 x plus 1 right parenthesis end fraction,
    Step 2 of 2:
    Now, find the product of fraction numerator x squared minus x cubed plus 4 x plus 11 over denominator left parenthesis x plus 3 right parenthesis left parenthesis 2 x plus 1 right parenthesis end fraction space cross times space fraction numerator 2 x squared plus 7 x plus 3 over denominator x plus 2 end fraction text .  end text
    For that, simplify and cancel out the common factors
    fraction numerator x squared minus x cubed plus 4 x plus 11 over denominator left parenthesis x plus 3 right parenthesis left parenthesis 2 x plus 1 right parenthesis end fraction space cross times space fraction numerator 2 x squared plus 7 x plus 3 over denominator x plus 2 end fraction equals fraction numerator x squared minus x cubed plus 4 x plus 11 over denominator left parenthesis x plus 3 right parenthesis left parenthesis 2 x plus 1 right parenthesis end fraction space cross times space fraction numerator 2 x squared plus 6 x plus x plus 3 over denominator x plus 2 end fraction
    equals fraction numerator x squared minus x cubed plus 4 x plus 11 over denominator left parenthesis x plus 3 right parenthesis left parenthesis 2 x plus 1 right parenthesis end fraction space cross times space fraction numerator 2 x left parenthesis x plus 3 right parenthesis plus 1 left parenthesis x plus 3 right parenthesis over denominator x plus 2 end fraction
    equals fraction numerator x squared minus x cubed plus 4 x plus 11 over denominator left parenthesis x plus 3 right parenthesis left parenthesis 2 x plus 1 right parenthesis end fraction space cross times space fraction numerator left parenthesis 2 x plus 1 right parenthesis left parenthesis x plus 3 right parenthesis over denominator left parenthesis x plus 2 right parenthesis end fraction
    equals fraction numerator open parentheses x squared minus x cubed plus 4 x plus 11 close parentheses over denominator left parenthesis x plus 2 right parenthesis end fraction
    Hence, the simplified expression is fraction numerator open parentheses x squared minus x cubed plus 4 x plus 11 close parentheses over denominator left parenthesis x plus 2 right parenthesis end fraction;

    LCM of two numbers or values are the least common multiples of both of them.

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