Divide using the long division method and check the answer.
\[ - 11x + 5x - 4\]by \[2x - 1\]

Hint:Here we are asked to divide the given algebraic expression by another algebraic expression by the long division method. First, we will try to simplify the given expression which is going to get divided if there is any possibility to simplify. Then we will start dividing it by the long division method. Here we are also asked to check whether the answer is correct or not for that, we will use the division algorithm and check our answer.

Complete step-by-step solution:
We aim to divide the algebraic expression \[ - 11x + 5x - 4\] by another algebraic expression \[2x - 1\].
We can see that the expression \[ - 11x + 5x - 4\] can be simplified further. Let’s simplify it by grouping the like terms.
\[ - 11x + 5x - 4\]\[ = ( - 11x + 5x) - 4\]
Thus, we get \[ - 6x - 4\]
Now let us start to divide the expression\[ - 6x - 4\] by \[2x - 1\] using the long division method.
We first aim to eliminate the first term in the expression \[ - 6x - 4\] (that is \[ - 6x\]). For that, we need to find a number or an expression which will give the first term (that is \[ - 6x\]) when multiplied with the expression \[2x - 1\].
On multiplying the expression \[2x - 1\] by \[ - 3\] we get \[ - 6x + 3\]. Here we got the first term on multiplying \[2x - 1\] by \[ - 3\] which will be easier for us to eliminate that term.
\[2x - 1\mathop{\left){\vphantom{1
   - 6x - 4 \\
  \underline { - 6x + 3} \\
 }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{
   - 6x - 4 \\
  \underline { - 6x + 3} \\
 }}}
\limits^{\displaystyle \,\,\, { - 3}}\]
Now let us subtract to get the remainder.
\[2x - 1\mathop{\left){\vphantom{1
  {\text{ }} - 6x - 4 \\
  \underline { - \left( { - 6x + 3} \right)} \\
 }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{
  {\text{ }} - 6x - 4 \\
  \underline { - \left( { - 6x + 3} \right)} \\
 }}}
\limits^{\displaystyle \,\,\, { - 3}}\]
Let us multiply the minus sign inside and then subtract it.
\[2x - 1\mathop{\left){\vphantom{1
   - 6x - 4 \\
  {\text{ }}\underline {6x - 3} \\
  {\text{ }} - 7 \\
 }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{
   - 6x - 4 \\
  {\text{ }}\underline {6x - 3} \\
  {\text{ }} - 7 \\
 }}}
\limits^{\displaystyle \,\,\, { - 3}}\]
Now we cannot proceed with the long division method as we got a constant as a remainder.
Thus, on dividing \[ - 11x + 5x - 4\] by \[2x - 1\] we get \[ - 3\] as quotient and \[ - 7\] as remainder.
Now let us check whether the answer we got is correct or not.
We know that by division algorithm, the dividend will be equal to the product of divisor and quotient plus a remainder. That is the dividend \[ = \]divisor\[ \times \] quotient\[ + \] remainder.
Here, the dividend is \[ - 6x - 4\], the divisor is \[2x - 1\], the quotient is \[ - 3\], and the remainder is \[ - 7\]. Thus, we get
\[ - 6x - 4 = (2x - 1) \times ( - 3) + ( - 7)\]
\[ - 6x - 4 = ( - 6x + 3) + ( - 7)\]
\[ - 6x - 4 = - 6x + 3 - 7\]
\[ - 6x - 4 = - 6x - 4\]
Thus, we verified that our answer is correct. Therefore, on dividing \[ - 11x + 5x - 4\] by \[2x - 1\] we get \[ - 3\] as quotient and \[ - 7\] as remainder.

Note:In this problem, we have divided the given algebraic equation using the long division method. We also need to check the answer we found by using the division algorithm. Also, it was necessary to simplify the given expression as it made our calculation easier.