Answer: The quotient of the rational expression is 3.
Step-by-step explanation:
since we have given that
[tex]\frac{\frac{x^2+5x+4}{x-1}}{\frac{x^2-16}{3x-3}}[/tex]
We need to solve for the quotient ;
We will change it in reduced form:
[tex]\frac{\frac{x^2+5x+4}{x-1}}{\frac{x^2-16}{3x-3}}\\\\=\frac{(x^2+5x+4)\times (3x-3)}{(x^2-16)\times (x-1)}[/tex]
First we solve the quadratic equation by using "Split the middle term":
[tex]x^2+5x+4\\\\=x^2+4x+x+4\\\\=x(x+4)+1(x+4)\\\\=(x+4)(x+1)[/tex]
Now, put the factorised form in the above expression:
[tex]\frac{(x^2+5x+4)\times (3x-3)}{(x^2-16)\times (x-1)}\\\\=\frac{(x+4)(x+1)\times 3(x-1)}{(x+4)(x-4)(x-1)}\\\\=\frac{3(x+1)}{x-4}\\\\=\frac{3x+3}{x-4}\\\\=3+\frac{15}{x-4}[/tex]
Hence, the quotient of the rational expression is 3.
