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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.

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)

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)