Since there is more than one term in the denominator, it is not straightforward to see what can be divided out here. So we first want to factor the numerator and denominator.

In this case, the numerator has a common factor of x and the denominator has a common factor of 3. Factoring these out gives:\frac{2x^2 + 7x}{6x + 21} = \frac{x\left(2x + 7\right)}{3\left(2x + 7\right)}We can now see the common factor of 2x + 7 which can be divided out from the numerator and denominator:\frac{x\left(2x + 7\right)}{3\left(2x + 7\right)} = \frac{x}{3}

So the simplified form is \dfrac{x}{3}.

When simplifying a rational expression, the difference in the degrees of the numerator and denominator does not change. We can use this to help check if we have incorrectly factored or divided somewhere (but note that having this difference be the same doesn't check for other types of errors).

Since there is more than one term in the denominator, it is not straightforward to see what can be divided out here. So we first want to factor the numerator and denominator.

In this case, the numerator is a quadratic. To factor it, we want two numbers with a sum of 5 and a product of -24, and these two numbers are 8 and -3.

The denominator is a difference of two cubes, since 27 = 3^3, so we can factor it using that property. Factoring the two expressions gives:\frac{y^2 + 5y - 24}{y^3 - 27} = \frac{\left(y + 8\right)\left(y - 3\right)}{\left(y - 3\right)\left(y^2 + 3y + 9\right)}We can now see the common factor of y - 3 which can be divided out from the numerator and denominator:\frac{\left(y + 8\right)\left(y - 3\right)}{\left(y - 3\right)\left(y^2 + 3y + 9\right)} = \frac{y + 8}{y^2 + 3y + 9}

So the simplified form is \dfrac{y + 8}{y^2 + 3y + 9}.

In this case, the simplified form appears to have about the same "complexity" as the original expression, since it has the same number of terms between the numerator and denominator as the original expression did.

The reason that this form is simpler is that the exponents involved are smaller. The numerator and denominator of the original expression had degrees of 2 and 3 respectively, while the simplified expression has degrees of 1 and 2 respectively.