How do you factor the polynomials $ 48tu - 90t + 32u - 60 $ ?

Hint: Factoring a polynomial means finding factors of the given polynomial and converting the given expression as product of the factors. A factor is a number or an expression which divides the given expression completely. A polynomial can have no factors or any number of factors.

Complete step by step solution:
We have to factor the given polynomial $ 48tu - 90t + 32u - 60 $ .
We can observe that this is a four-term polynomial with $ t $ and $ u $ as variables.
To factor the polynomial means writing the given expression as a product of the factors. So first we try to find the factors of the polynomial.
To find the factor we group the first two terms together and the last two terms together, as shown,
 $ \left( {48tu - 90t} \right) + \left( {32u - 60} \right) $
Now in the first group we can observe that we can write it as,
 $ \left( {48tu - 90t} \right) = \left( {8u \times 6t} \right) - \left( {15 \times 6t} \right) $
From distributive property of multiplication, which states that $ \left( {a + b} \right) \times c = \left( {a \times c} \right) + \left( {b \times c} \right) $ , we can take $ 6t $ common from both the terms and write the above expression as,
 $ \left( {8u \times 6t} \right) - \left( {15 \times 6t} \right) = 6t\left( {8u - 15} \right) $
Observe that $ 6t $ is the HCF of both the terms in $ \left( {48tu - 90t} \right) $ .
Similarly we can simplify the second group $ \left( {32u - 60} \right) $ .
On observing that $ 4 $ is the HCF of both terms, we can write it as,
 $ \left( {32u - 60} \right) = \left( {8u \times 4} \right) - \left( {15 \times 4} \right) = 4\left( {8u - 15} \right) $
Thus, our expression becomes,
 $
  \left( {48tu - 90t} \right) + \left( {32u - 60} \right) \\
   = 6t\left( {8u - 15} \right) + 4\left( {8u - 15} \right) \;
  $
Again using the distributive property of multiplication we can simplify this further as follows,
 $
  6t\left( {8u - 15} \right) + 4\left( {8u - 15} \right) \\
   = \left( {6t + 4} \right)\left( {8u - 15} \right) \;
  $
We observe that we get the result as a product of two expressions. This is our final result where each of both the expressions $ \left( {6t + 4} \right) $ and $ \left( {8u - 15} \right) $ are the factors of the given polynomial.
Hence, $ 48tu - 90t + 32u - 60 = \left( {6t + 4} \right)\left( {8u - 15} \right) $
So, the correct answer is “ $ \left( {6t + 4} \right)\left( {8u - 15} \right) $ ”.

Note: A polynomial can have no factor or many factors. Factoring the polynomial means writing it in the form of a product of two or more expressions which are known as its factors. Also, $ 1 $ is a common factor for all the expressions but we don’t write it explicitly in the results. We can check our result by multiplying the factors and arriving at the given polynomial.