Discussion
If \(15x^{2} - 26x + 8 = (Ax + B)(Cx + D)\), where A and C are positive integers, then what is the value of \((2A + B - C - 2D)\) ?
Answer: B
The value of \((2A + B - C - 2D)\) is 1.
Split the middle term: the product \(15 \times 8 = 120\) has the factor pair \(-6\) and \(-20\), which add to \(-26\).
\[ \begin{aligned} 15x^{2} - 26x + 8 &= 15x^{2} - 6x - 20x + 8 \\ &= 3x(5x - 2) - 4(5x - 2) \\ &= (3x - 4)(5x - 2) \end{aligned} \]
Taking \(A = 3, B = -4, C = 5, D = -2\):
\[ 2(3) + (-4) - 5 - 2(-2) = 6 - 4 - 5 + 4 = 1 \]
Writing the factors the other way round (\(A = 5, B = -2, C = 3, D = -4\)) gives 13, which is not offered, so the intended factorisation is the one above. 0, 2 and 3 come from sign slips with the negative B and D.
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