how to factor x^2 + 3x -54

Question

Can't figure this out from the book.  Answer in book is (x+9)(x-6)  Directions are not clear

Answer 1

you have a quadratic expression in the form ax2 + bx + c where a = 1
You can always use the quadratic formula to find the zeros or roots of such a function, but there is also a process whereby we factor the expression into two binomials which have the form

 (x + m) (x + n)

We find the value of m and n as follows:
Start with the constant c. We need two numbers m and n that multiply to give the product c and also add to give the sum b. 
Mathematically we write    mn = c     m + n = b

In your example, c= -54. The sets of factors of -54 are 1 x -54, 2 x -27, 3 x -18, 6 x -9. And of course we can switch the negative sign in each case and get 4 more choices.
Now we have the choices for mn = -54
Which set has a sum of +3? The answer is 9 and -6
Finally we get our binomial factors which are     (x + 9) (x - 6)
You can check the answer by multiplying the factors to see if you get the original quadratic polynomial.

Answer 2

Factor -54. It is -9*6 or 9*(-6). Then check the sum of factors, it shall come to -3, opposite to the coefficient in front of x. -9+6=-3, so roots are x₁=-9 and x₂=6. The factoring is given by:

 

(x-x₁)(x-x₂), in your case, 

(x+9)(x-6)