(5sqrt(2x+5)-1)/(sqrt(2x+5)+2)

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Identify Expression: Identify the expression to be simplified. We have the expression (5sqrt(2x+5)-1)/(sqrt(2x+5)+2). Our goal is to simplify this expression.Rationalize Denominator: Rationalize the denominator. To simplify the expression, we need to get rid of the square root in the denominator. We can do this by multiplying the numerator and the denominator by the conjugate of the denominator, which is (sqrt(2x+5)-2).Multiply Numerator: Multiply the numerator by the conjugate of the denominator. (5sqrt(2x+5)-1) * (sqrt(2x+5)-2)Apply Distributive Property: Apply the distributive property (FOIL method) to the numerator. (5sqrt(2x+5)-1) * (sqrt(2x+5)-2) = 5(2x+5) - 10sqrt(2x+5) + sqrt(2x+5) - 2Combine Like Terms: Combine like terms in the numerator. 5(2x+5) - 10sqrt(2x+5) + sqrt(2x+5) - 2 = 10x + 25 - 9sqrt(2x+5) - 2Simplify Numerator: Simplify the terms in the numerator. 10x + 25 - 9sqrt(2x+5) - 2 = 10x + 23 - 9sqrt(2x+5)Multiply Denominator: Multiply the denominator by the conjugate of the denominator. (sqrt(2x+5)+2) * (sqrt(2x+5)-2)Apply Difference of Squares: Apply the difference of squares formula to the denominator. (sqrt(2x+5)+2) * (sqrt(2x+5)-2) = (sqrt(2x+5))^2 - (2)^2Simplify Denominator: Simplify the denominator. (sqrt(2x+5))^2 - (2)^2 = 2x+5 - 4Write Simplified Expression: Combine like terms in the denominator. 2x+5 - 4 = 2x + 1Write the simplified expression. The simplified expression is (10x + 23 - 9sqrt(2x+5)) / (2x + 1).

$\frac{5 \sqrt{2 x+5}-1}{\sqrt{2 x+5}+2}$

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$\frac{5 \sqrt{2 x+5}-1}{\sqrt{2 x+5}+2}$