Question

what is the quotient?
\\(\frac{t + 3}{t + 4} \div (t^2 + 7t + 12)\\)
\\(\bigcirc\\ (t + 3)^2\\)
\\(\bigodot\\ (t + 4)^2\\)
\\(\bigcirc\\ \frac{1}{(t + 4)^2}\\)
\\(\bigcirc\\ \frac{1}{(t + 3)^2}\\)

Explanation:

Step1: Rewrite division as multiplication

To divide by a polynomial, we multiply by its reciprocal. So, \(\frac{t + 3}{t + 4}\div(t^{2}+7t + 12)=\frac{t + 3}{t + 4}\times\frac{1}{t^{2}+7t + 12}\)

Step2: Factor the quadratic

Factor \(t^{2}+7t + 12\). We need two numbers that multiply to \(12\) and add to \(7\), which are \(3\) and \(4\). So, \(t^{2}+7t + 12=(t + 3)(t + 4)\)

Step3: Substitute and simplify

Substitute the factored form into the expression: \(\frac{t + 3}{t + 4}\times\frac{1}{(t + 3)(t + 4)}\)
Cancel out the common factor \((t + 3)\) from the numerator and denominator: \(\frac{1}{(t + 4)}\times\frac{1}{(t + 4)}=\frac{1}{(t + 4)^{2}}\)

Answer:

\(\frac{1}{(t + 4)^{2}}\) (the third option: \(\frac{1}{(t + 4)^{2}}\))