Add the following rational expressions.
$\frac{3x+5}{x+2} + \frac{4x-2}{2x+4} = \frac{\square}{2x+4}$
To solve the problem, we need to add the two given rational expressions:
\[\frac{3x+5}{x+2} + \frac{4x-2}{2x+4} = \frac{\square}{2x+4}.\]
Our objective is to determine the numerator that completes the expression on the right-hand side. The key to adding rational expressions is to ensure they have a common denominator. We observe that the denominator of the second fraction, \(2x+4\), can be factored as \(2(x+2)\), which contains the denominator of the first fraction, \(x+2\). Thus, the common denominator for the sum is \(2x+4\).
To write the first fraction \(\frac{3x+5}{x+2}\) with the common denominator \(2x+4\), we multiply both its numerator and denominator by 2:
\[\frac{3x+5}{x+2} = \frac{2 \times (3x+5)}{2 \times (x+2)} = \frac{6x+10}{2x+4}.\]
Now, the expression becomes:
\[\frac{6x+10}{2x+4} + \frac{4x-2}{2x+4}.\]
Adding the two fractions with the same denominator, we combine their numerators:
\[\frac{(6x+10) + (4x-2)}{2x+4}.\]
Combining like terms in the numerator, we get:
\[(6x+4x) + (10-2) = 10x+8.\]
Therefore, the sum is:
\[\frac{10x+8}{2x+4}.\]
The missing expression in the numerator is \(10x+8\).
Rational expressions are commonly used in physics to calculate the combined effect of forces, resistances, or velocities. For instance, if you are calculating the total resistance of two resistors in parallel, you add their reciprocals. Similarly, if you are modeling the average speed of an object moving over two different segments of a path, you often work with rational expressions. Mastering this algebraic skill allows you to simplify complex physical models and solve for unknown parameters efficiently, which is a fundamental tool for engineers and scientists.