 What about trying to work with rational expressions that have quadratic denominators? So let's say we want to simplify the sum of these two rational expressions. While we could find a common denominator by multiplying the denominators together, eventually we'd need to factor the denominators, so let's factor first. So for this first denominator, x squared plus 5x plus 4, we want this to be the product of x plus a times x plus b, so we need to try every pair of numbers a and b, where a b is equal to 4, and hope we get the middle term correct. Well, two numbers that multiply to 4 are 1 and 4, and so maybe x squared plus 5x plus 4 is x plus 1 times x plus 4. We have to check, so we'll expand, and sometimes we get lucky. Now remember factored form is best, so let's rewrite this fraction with a denominator in factored form. We also want to factor x squared plus x minus 12, and again, we hope that it's equal to x plus a times x plus b, and so we need to try every pair of numbers that multiplies to minus 12. So 1 and negative 12 multiply to minus 12, but before we sign the contract that says x squared plus x minus 12 is really equal to x plus 1 times x minus 12, we should check it out, and it's not true. So another pair of numbers that multiply to minus 12 are 2 and negative 6. We'll check it, and we won't sign this contract either. This is not a factorization. How about 3 and negative 4? Nope, but we are close. We have a minus x instead of a plus x, so that suggests if we change the order, x minus 3 times x plus 4, we do get a factorization, and so we'll write our second rational expression with the denominator in factored form. Now we see that both denominators have a common factor of x plus 4, and so we can make a common factor by supplying the missing factors of x plus 1 and x minus 3. So this first rational expression, 2 over x plus 4 times x plus 1, the missing factor is x minus 3, so we'll multiply the numerator and denominator by x minus 3 to get, and likewise, this second rational expression already has an x plus 4 and an x minus 3 in the denominator, so it's missing an x plus 1, so we'll multiply numerator and denominator by x plus 1, and now both rational expressions have the same denominator. And since they have the same denominator, we can combine the numerators and try to remove common factors. But remember, you can only remove common factors if you have a product. The denominator is a product, but the numerator is not. Since the numerator isn't a product, we can expand it, so we'll take our numerator to x minus 3 plus 5 x plus 1, we'll expand, and simplify. So a useful thing to remember at this point is that a factor only matters if it's a common factor. Since we left our denominator in factored form, we know that the only factors that matter in the numerator are x minus 3, x plus 4, or x plus 1. And we might take a look. We see that 7x minus 1 can't be a times x plus 1. And 7x minus 1 can't be something times x plus 4. And 7x minus 1 can't be something times x minus 3. And what this means is that there's no chance that 7x minus 1 will have a factor in common with the denominator, so we don't care how or even if it factors, we can leave our answer in this form.