Question

Simplify the expression. $$\frac{x+6}{x+1}-\frac{4}{2 x+3}$$

Short answer

The simplified form of the given expression is \(\frac{2x^{2}+5x+14}{(x+1)(2x+3)}.\)

Step-by-step solution

Find the Least Common Denominator (LCD)

The least common denominator (LCD) between \(x + 1\) and \(2x + 3\) is their product, which is \((x + 1)(2x + 3).\)

Rewrite the fractions with the LCD

Now, rewrite each fraction with the LCD as the denominator. The first fraction is multiplied by \(\frac{2x + 3}{2x + 3}\) and the second is multiplied by \(\frac{x + 1}{x + 1}\), resulting in \[\frac{x * (2x + 3) + 6 * (2x + 3)}{(x + 1)(2x + 3)} - \frac{4 * (x + 1)}{(x + 1)(2x + 3)}.\]

Simplify the numerator of the combined fraction

Simplify the expression in the numerator to obtain a single fraction. So, it becomes \(\frac{2x^{2}+9x+18-4x-4}{(x + 1)(2x + 3)}\). Simplify the expression in the numerator further, \(\frac{2x^{2}+5x+14}{(x + 1)(2x + 3)}\).

Attempt to factor and simplify the fraction

The next step is to factor the numerator and the denominator to see if there are common factors that can lead to more simplification. In this case, the numerator cannot be factored any further and the denominator is already in its most factored form.