If 15x² − 26x + 8 = (Ax + B) (Cx + D), where A and C are positive integers, then what is the value of (2A + B − C − 2D) ?

The correct answer is 1.

We are given the quadratic equation:

$15x^2-26x+8=(Ax+B)(Cx+D)$

To factorize the left side of the equation, we find two numbers whose product is $15\times 8=120$ and sum is $-26$. These numbers are $-20$ and $-6$.

Rewriting the middle term:

$15x^2-20x-6x+8$

Factorizing by grouping:

$5x(3x-4)-2(3x-4)$

$=(3x-4)(5x-2)$

Comparing this with $(Ax+B)(Cx+D)$, where $A$ and $C$ are positive integers, we set:

$A=3$, $B=-4$, $C=5$, and $D=-2$

Now, substituting these values into the expression $(2A+B-C-2D)$:

$2A+B-C-2D=2\left(3\right)+(-4)-5-2(-2)$

$=6-4-5+4=1$

Hence, the value of $(2A+B-C-2D)$ is 1.