Simplifying a rational expression involving polynomials; so whenever we are dealing a rational expression and we want to simplify, we look for common factors in both the numerator and the denominator. When we are dealing with polynomials it’s not different. We could go ahead and multiply this out but what we are going to end up with is a pretty ugly multiplication process and a pretty ugly polynomial in both our numerator and our denominator. By factoring out each term individually we are going to be able to find our common factors much easier.
So just going down the line and factoring each polynomial, what we end up with is x plus 2 times x plus 1 for the first numerator, this factors to x plus 3 x plus 1. x² minus 4 is just the difference of squares so that’s x minus 2 x plus 2 and lastly x² plus 7x plus 12, 3 and 4 so this turns into x plus 3 times x plus 4.
So what we did is we basically factored every single quadratic we have and now we can just factor or cancel out any factors that we have in common. So we have a x plus 3 and an x plus 3, a x plus 1 and an x plus 1, and an x plus 2, and an x plus 2. Leaving us with the only things behind being an x plus 4 in the numerator and an x minus 2 in the denominator.
So multiplying rational expressions involving polynomials just factor them out and then cancel your factors and it becomes pretty straight forward.