Question

2x square+3x-152 factorize it

Answer 1

2x²+3x-152
=2x²-16x+19x-152
=2x(x-8)+19(x-8)
=(2x+19)(x-8)

Answer 2

There are many methods.  One is by just looking at the polynomial expression. By guessing. Another is by finding rational factors like below.  Another is by the solution for P(x)=0, roots of a quadratic equation.  [-b+-\sqrt(b^2-4ac)] /2.   I state one of these.

$P(x)=2x^2+3x-152\\ \\Let\ us\ say\ there\ are \ rational\ factors.\\ \\the rational\ factors\ are\ possibly\ among\ the\ following:\\ \\+- \frac{a\ factor\ of\ 152}{a\ factor\ of\ 2}\\ \\+-1, +- 1/2, +-4/2, +-4,+-8/2,+-8,\\+-19/2, +-19,+-38,+-76,+-152$

Try x =1,  2*1²+3*1-152 = -145    very high value.
So try x = 4,  2*4² + 3 * 4 - 152 = -108   still negative and hig h value
so try x = 8,  2 * 8² + 3 * 8 - 152 = 128+24-152 = 0
so x = 8 is a factor.
    Then :  (- 152 / 2 ) / 8   is the other factor  = -19/2

Factorize by looking at the coefficient of x² and the constant term -152.

(x - 8) (2x + 152/8) = (x-8)(2x +19)