Full solution

Q. Simplify the following:

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\frac{2}{x+2}-\frac{4}{2x+1}

Identify Terms and Denominators: Identify the terms and their denominators. We have two fractions with different denominators: $\frac{2}{x+2}$ and $\frac{4}{2x+1}$ .
Find Common Denominator: Find a common denominator for the two fractions. The common denominator will be the product of the two distinct denominators, which is x+\(2)( $2$ x+ $1$ )\.
Rewrite with Common Denominator: Rewrite each fraction with the common denominator. The first fraction becomes $(2)(2x+1)/[(x+2)(2x+1)]$ , and the second fraction becomes $(4)(x+2)/[(x+2)(2x+1)]$ .
Expand Numerators: Expand the numerators of both fractions. For the first fraction, $(2)(2x+1)$ becomes $4x+2$ . For the second fraction, $(4)(x+2)$ becomes $4x+8$ .
Combine Fractions: Combine the two fractions by subtracting the second fraction from the first, using the common denominator. This gives us $(4x+2 - (4x+8))/[(x+2)(2x+1)]$ .
Simplify Numerator: Simplify the numerator by combining like terms. Subtracting $4x+8$ from $4x+2$ gives us $-6$ .
Write Final Expression: Write the simplified expression. The final simplified expression is $(-6)/[(x+2)(2x+1)]$ .