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Factor the trinomial

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You're being asked to factor, not solve. If you can't factor, then you declare it prime. While you need to know the set of numbers to consider it prime, most often in these algebra questions, they restrict the coefficients to INTEGERS. If you're not certain, check with your teacher. So binomials of (x+1/2)(x+3) would suggest prime because the x^0 coefficient in the first factor is 1/2=0.5 which is not an integer.

For this reason, you should use caution if you choose methods like completing the square or the quadratic formula. You can use them - but you need to come back to the factors and see if all coefficients are integer. (Again, prime or factorable over the set of integers is assumed in this case.) 

So start with factoring methods!

Well, 5·4 is 20 which is one away from 19.  4 and 1 are factors of 4....let's do this! (Some teachers may formally call this the AC Method since we're looking at coefficients 'a' and 'c'.)

(5x     )(x      )  -- This will give us 5x^2 +-..

(5x    1)(x    4)  -- This can give us 20x and 1x. 19 is one less than 20 and c= -4, so yeah this will work.

(5x - 1)(x + 4)  -- Check it: 5x^2 + (20x-x) -4 = 5x^2 +19x -4

Just for extra practice, you could solve it and check your factors. Set each factor to 0 because by the Zero Property of Multiplication, if either factor is 0, the product is 0. You should get: x = 1/5, -4   Plugging either of these into the original polynomial gives you 0, so your factors are correct!

Remember though....just the factors! (a) 5x^2 + 19x -4 = (5x - 1)(x + 4)