Question

consider the rational expression shown below:
\\(\frac{x - 4}{x + 3}-\frac{3x + 2}{x^{2}+7x + 12}\\)
rewrite as the addition of two fractions with least common denominators. leave the denominators in factored form.
left fraction + right fraction
combine all terms and write as a single fraction. do not cancel any common factors.
completely simplify the expression:

Explanation:

Step1: Factor the second - denominator

Factor \(x^{2}+7x + 12=(x + 3)(x+4)\).

Step2: Find the least - common denominator (LCD)

The first fraction has a denominator of \(x + 3\) and the second has a denominator of \((x + 3)(x + 4)\). The LCD is \((x + 3)(x + 4)\).

Step3: Rewrite the first fraction with the LCD

Multiply the numerator and denominator of \(\frac{x - 4}{x + 3}\) by \(x + 4\): \(\frac{(x - 4)(x + 4)}{(x + 3)(x + 4)}=\frac{x^{2}-16}{(x + 3)(x + 4)}\).
The second fraction is \(-\frac{3x + 2}{(x + 3)(x + 4)}\).
Rewrite as addition: \(\frac{x^{2}-16}{(x + 3)(x + 4)}+\frac{-(3x + 2)}{(x + 3)(x + 4)}\)
LEFT fraction: \(\frac{x^{2}-16}{(x + 3)(x + 4)}\)
RIGHT fraction: \(\frac{-(3x + 2)}{(x + 3)(x + 4)}\)

Step4: Combine the fractions

\(\frac{x^{2}-16-(3x + 2)}{(x + 3)(x + 4)}=\frac{x^{2}-16-3x - 2}{(x + 3)(x + 4)}=\frac{x^{2}-3x-18}{(x + 3)(x + 4)}\)

Step5: Simplify the numerator

Factor the numerator \(x^{2}-3x - 18=(x-6)(x + 3)\).
So the simplified fraction is \(\frac{x-6}{x + 4}\)

Answer:

LEFT fraction: \(\frac{x^{2}-16}{(x + 3)(x + 4)}\)
RIGHT fraction: \(\frac{-(3x + 2)}{(x + 3)(x + 4)}\)
Combined single - fraction: \(\frac{x^{2}-3x-18}{(x + 3)(x + 4)}\)
Simplified expression: \(\frac{x-6}{x + 4}\)