To simplify rational expressions, it is often helpful to factor all of the involved polynomials:
10/(x^2-16x+60) + 7/(x^2-100)
= 10 /[(x-6)(x-10)] + 7/[(x-10)(x+10)]
Since both terms in the expression have an (x-10) in the denominators, we only need an extra (x-6) in the first term and an (x+10) in the second. Now, we cannot willy-nilly add terms to products, but we can multiply expressions by 1 and preserve them.
10(x+10) /[(x-6)(x-10)(x+10)] + 7(x-6)/[(x-10)(x+10)(x-6)]
= (10x+100)/[~] + (7x-42)/[~] (I am using ~ for the common denominator.)
= (10x+100 + 7x-42)/[~] (We can add two fractions since we have a common denominator.)
For your second expression, please verify the first denominator as I presume you added an x in there.