Final answer:
To simplify the expression, we factor the denominators and find a common denominator. Then, we combine the numerators over the common denominator and simplify the expression.
Step-by-step explanation:
To simplify the expression $rac{3x^2}{x-7} - rac{2x}{x^2-5x-14}$, we first factor the denominators. The denominator of the first fraction is a binomial $x-7$, which cannot be factored further. The denominator of the second fraction is a trinomial $x^2-5x-14$, which can be factored as $(x-7)(x+2)$. We can now write the expression as $rac{3x^2}{x-7} - rac{2x}{(x-7)(x+2)}$.
To add or subtract fractions with different denominators, we need to find a common denominator. In this case, the common denominator is $(x-7)(x+2)$. To get the first fraction to have this denominator, we need to multiply the numerator and denominator by $(x+2)$. To get the second fraction to have this denominator, we need to multiply the numerator and denominator by $(x-7)$. After performing these operations, we get the expression $rac{3x^2(x+2)}{(x-7)(x+2)} - rac{2x(x-7)}{(x-7)(x+2)}$.
Now, we can combine the numerators over the common denominator, which gives us $rac{3x^2(x+2) - 2x(x-7)}{(x-7)(x+2)}$. Expanding the parentheses, we have $rac{3x^3 + 6x^2 - 2x^2 + 14x}{(x-7)(x+2)}$.
Simplifying the numerator further, we get $rac{3x^3 + 4x^2 + 14x}{(x-7)(x+2)}$. This is the simplified form of the expression.