How to factor things with a coefficient that isn't one

An easy way than guessing and checking? There was some little trick for it, something with multiplying a & c...?

for example: 2x^2 + 5x -12?

what happens when there aren't three terms like

4x^2 - 25


thanks in advance

For the first one, the best way to factor these in general is to factor by grouping.

The method is as follows: take the x^2 coefficient and multiply it by the constant term. Find two factors of that product that add up to the x coefficient. Using those two factors, split up the middle term into two using those factors, group the the four terms into two sets of two terms, factor out the GCFs from each set, and factor the common factor out.

With your example, the x^2 coefficient is 2 and the constant is -12. Their product is -24. So, we need two numbers that multiply to make -24 and add up to make 5 (8 and -3). Splitting up the middle term with these as coefficients gives:
2x^2 + 8x - 3x - 12.

Grouping the first two and last two terms:
(2x^2 + 8x) + (-3x - 12).

Factor out the GCFs from the two pairs of terms:
2x(x + 4) - 3(x + 4).

Factor out x + 4:
(x + 4)(2x - 3).

For the second one, you can factor this using a difference of two squares:
4x^2 - 25 = (2x)^2 - 5^2
= (2x + 5)(2x - 5), since a^2 - b^2 = (a + b)(a - b).

I hope this helps!

The way I do it is, if i can factor the coefficient out, then i do. If not, then i look to see where the expression would equal 0. Then x minus that number would be one of the terms. From there just see what you would have to multiply that term by to get the original. For the example you have, the expression would equal 0 when x = -4. So (x+4) is one term. From there i see that i need a 2x^2, so i would have to multiple the first term by 2x. So far we have (x+4)(2x -/+ __). Then we see we need a -12, and we have a +4, so +4 x -3 = -12. So factored, the expression is (x+4)(2x-3)

The special cases are

a^2-b^2 = (a-b)(a+b)

(a-b)^2 = (a^2-2ab+b^2)

(a+b)^2 = (a^2+2ab+b^2)

(a-b)^3 = (a-b)(a^2+ab+b^2)

You should memorize these special cases and when you start to do problems you will see them easier and do them quicker.

(a+b)^3 = (a+b)(a^2-ab+b^2)

Multiply the leading coefficient with the constant term,
2*(-12) = -3*8
Attn: -3x+8x = 5x
Therefore, to get 8x, you need 2x*4.
(2x + ? )(? + 4)
Now fill in the two ? positions,
(2x-3)(x+4)