Your factoring is correct.
1 is not a prime number.
But the question "is it prime" has nothing to do with 5 and 3 or 5 and 1.
You cannot factor a prime polynomial. You'd have to use the quadratic formula.
The fact that you were able to factor it proves it isn't prime.
The test for prime is step 1. If it's prime, use the quadratic.
Remember the general quadratic polynomial is 
.
To test for prime determine if 
, that's called the discriminant, is a perfect square. If it is, then the polynomial isn't prime and you can factor it.
In this case the discriminant is 
4900 is a perfect square, so the polynomial is not prime.
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