Simplify. Express your answer as a single fraction in simplest form \( \frac{4 u}{2 u+4}+\frac{5 u}{10 u+4} \)
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To simplify the expression \( \frac{4 u}{2 u+4}+\frac{5 u}{10 u+4} \), we first simplify each fraction. The first fraction: \[ \frac{4u}{2u + 4} = \frac{4u}{2(u + 2)} = \frac{2u}{u + 2} \] The second fraction: \[ \frac{5u}{10u + 4} = \frac{5u}{2(5u + 2)} = \frac{5u}{2(5u + 2)} \] Now, we need a common denominator to combine these fractions. The least common denominator (LCD) of \( 2(u + 2) \) and \( 2(5u + 2) \) is \( 2(u + 2)(5u + 2) \). Now, rewrite both fractions with the common denominator: \[ \frac{2u}{u + 2} = \frac{2u \cdot 5u + 2}{(u + 2)(5u + 2)} = \frac{10u^2 + 4u}{2(u + 2)(5u + 2)} \] \[ \frac{5u}{2(5u + 2)} = \frac{5u(u + 2)}{2(u + 2)(5u + 2)} = \frac{5u^2 + 10u}{2(u + 2)(5u + 2)} \] Now combine them: \[ \frac{10u^2 + 4u + 5u^2 + 10u}{2(u + 2)(5u + 2)} = \frac{15u^2 + 14u}{2(u + 2)(5u + 2)} \] We can factor out a common factor from the numerator: \[ \frac{u(15u + 14)}{2(u + 2)(5u + 2)} \] Now the expression is simplified, and we cannot simplify it further since \( u(15u + 14) \) does not factor in a way that matches the denominator. Thus, the final answer is: \[ \frac{u(15u + 14)}{2(u + 2)(5u + 2)} \]
