what is the least common denominator of the rational expressions below?6/x^2+7x - 8/x^2+10x+21a. x(x+7)(x+3)b. x+7c. x(x+7)^2(x+3)d. x(x+3)

The first thing we need to do is factor the denominators in the rational expressions.
Notice that in the denominator of the first rational expression we have a common factor [tex]x[/tex], so we can factor x out:
[tex] x^{2} +7x=x(x+7)[/tex]
Now, the denominator of the second rational expression is a quadratic polynomial, we can factor it by finding tow numbers whose product will 21 and its sum will be 10. Those numbers are 7 and 3, (7x3=21 and 7+3=10):
[tex] x^{2} +10x+21=(x+7)(x+3)[/tex]

Lets rewrite our rational expression with our factored denominators:
[tex] \frac{6}{ x(x+7)} - \frac{8}{(x+7)(x+3)} [/tex]

Now, to find the least common denominator we are going to take one of the common factors (x+7) and all the non-common factors: x and (x+3):
[tex]x(x+7)(x+3)[/tex]

We can conclude that the correct answer is a. x(x+7)(x+3)