 First, always factor out the greatest common factor of all the terms.
 What is Factoring a Quadratic Trinomial?
 A quadratic trinomial (polynomial of degree 2 in one variable) has the form
ax^{2} + bx + c, a not 0.
 The quadratic trinomial is given.
 The goal is to find numbers A, B, C, and D so that
(Ax + B)(Cx + D) = ax^{2} + bx + c.
 Then (Ax + B)(Cx + D) is the factored form of ax^{2} + bx + c.
 Then (Ax + B) and (Cx + D) are the factors of ax^{2} + bx + c.
 How Are the Factors (Ax + B) and (Cx + D) Found?
 By trial and error, find numbers A, B, C, and D so that
A · C = a,
A · D + B · C = b, and
B · D = c

 This means that
 A and C must be factors of the coefficient a,
 B and D must be factors of the coefficient c, and
 A · D + B · C must equal the coefficient b.
 Why?
 Multiplying out (Ax + B)(Cx + D) gives that
ACx^{2} + (AD + BC)x + BD =

ax^{2} + bx + c.

 Equating coefficients on left and right hand sides gives
A · C = a,
A · D + B · C = b, and
B · D = c

 Steps to finding A, B, C, and D
 Choose A and C such that A · C = a.
 Choose B and D such that B · D = c.
 Repeat steps 1 and 2 until A · D + B · C = b.
 Not all polynomials can be factored. If you have exhausted all the possiblities for A, B, C, and D and been unable to factor the polynomial, then the polynomial cannot be factored.
 Factoring General Trinomials
 A general trinomial will have more than one variable. For example,
3x^{2} + 2xy  8y^{2}
 which factors into
 The goal when factoring general trinomials is to find expressions A, B, C, and D such that
(A + B)(C + D) equals the given polynomial.
 Note that A, B, C, and D may be expressions containing variables.
 Steps to finding A, B, C, and D
 Choose A and C such that A · C = the first term.
 Choose B and D such that B · D = last term.
 Repeat steps 1 and 2 until A · D + B · C = the middle term.
 Polynomials with Quadratic Form
 Sometimes, higher degree polynomials can be factored when observed to have quadratic form.
 A polynomial has quadratic form if the polynomial resulting from a variable substitution is quadratic.
 For example, y^{4} + 5y^{2} + 6 has quadratic form because the resulting polynomial is quadratic when x is substituted for y^{2}:

y^{4} + 5y^{2} + 6


=

(y^{2})^{2} + 5(y^{2}) + 6


=

(x)^{2} + 5(x) + 6

x is substituted for y^{2}

=

x^{2} + 5x + 6

resulting polynomial

 Then x^{2} + 5x + 6 factors into (x + 2)(x + 3).
 Finally, resubstituting y^{2} back in for x gives
y^{4} + 5y^{2} + 6 = (y^{2} + 2)(y^{2} + 3)
 Remember to check the two new factors to see if they can each be factored further.
 If One Factor of a Polynomial is Known....
 Use polynomial division (either synthetic or long) to find another factor (i.e., the quotient) of the polynomial.
 Try to factor the new factor (i.e., the quotient).
