Question
Factorise:$6\Big(2\text{x}-\frac{3}{\text{x}}\Big)^2+7\Big(2\text{x}-\frac{3}{\text{x}}\Big)-20$
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Answer
Given equation: $6\Big(2\text{x}-\frac{3}{\text{x}}\Big)^2+7\Big(2\text{x}-\frac{3}{\text{x}}\Big)-20$ Let $2\text{x}-\frac{3}{\text{x}}=\text{a}$
Then, we have $= 6a^2 + 7a - 20 = 6a^2 + 15a - 8a - 20$
$= 3a(2a + 5) - 4(2a + 5)$
$= (2a + 5)(3a - 4)$
$=\bigg[2\Big(2\text{x}-\frac{3}{\text{x}}\Big)+5\bigg]\bigg[3\Big(2\text{x}-\frac{3}{\text{x}}\Big)-4\bigg]$
$=\Big(4\text{x}-\frac{6}{\text{x}}+5\Big)\Big(6\text{x}-\frac{9}{\text{x}}-4\Big)$
Then, we have $= 6a^2 + 7a - 20 = 6a^2 + 15a - 8a - 20$
$= 3a(2a + 5) - 4(2a + 5)$
$= (2a + 5)(3a - 4)$
$=\bigg[2\Big(2\text{x}-\frac{3}{\text{x}}\Big)+5\bigg]\bigg[3\Big(2\text{x}-\frac{3}{\text{x}}\Big)-4\bigg]$
$=\Big(4\text{x}-\frac{6}{\text{x}}+5\Big)\Big(6\text{x}-\frac{9}{\text{x}}-4\Big)$
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