Question

How do you multiply $\left( 5x+4 \right)\left( 5x+5 \right)\left( 2x-6 \right)\left( x-4 \right)$?

Answer 1

Hint: Assume the given expression as ‘E’. Take 5 common from (5x + 5) and 2 common from (2x – 6) to simplify the expression. Now, select the first two terms and multiply them then consider the product with the third terms and at last with the fourth term in a serial wise manner to get the answer.

Complete step by step solution:
Here, we have been provided with the expression $\left( 5x+4 \right)\left( 5x+5 \right)\left( 2x-6 \right)\left( x-4 \right)$ and we are asked to multiply them. Let us assume this product as ‘E’, so we have,
$\Rightarrow E=\left( 5x+4 \right)\left( 5x+5 \right)\left( 2x-6 \right)\left( x-4 \right)$
We can clearly see that we have four linear expressions in x and each one of them contains two terms, so we can say that they are binomial expressions. Before multiplying we need to check if we can take anything common or not to reduce the calculation. We can take 5 common from (5x + 5) and 2 common from (2x – 6), so we have,
$\begin{align}
  & \Rightarrow E=5\times 2\left( 5x+4 \right)\left( x+1 \right)\left( x-3 \right)\left( x-4 \right) \\
 & \Rightarrow E=10\left( 5x+4 \right)\left( x+1 \right)\left( x-3 \right)\left( x-4 \right) \\
\end{align}$
Let us start with considering the product of first and second term, so we get,
$\begin{align}
  & \Rightarrow E=10\left( 5{{x}^{2}}+5x+4x+4 \right)\left( x-3 \right)\left( x-4 \right) \\
 & \Rightarrow E=10\left( 5{{x}^{2}}+9x+4 \right)\left( x-3 \right)\left( x-4 \right) \\
\end{align}$
Further multiplying with the third term we get,
$\begin{align}
  & \Rightarrow E=10\left( 5{{x}^{3}}-15{{x}^{2}}+9{{x}^{2}}-27x+4x-12 \right)\left( x-4 \right) \\
 & \Rightarrow E=10\left( 5{{x}^{3}}-6{{x}^{2}}-23x-12 \right)\left( x-4 \right) \\
\end{align}$
At last considering the product with the fourth term we get,
$\begin{align}
  & \Rightarrow E=10\left( 5{{x}^{4}}-20{{x}^{3}}-6{{x}^{3}}+24{{x}^{2}}-23{{x}^{2}}+92x-12x+48 \right) \\
 & \Rightarrow E=10\left( 5{{x}^{4}}-26{{x}^{3}}+{{x}^{2}}+80x+48 \right) \\
 & \therefore E=50{{x}^{4}}-260{{x}^{3}}+10{{x}^{2}}+800x+480 \\
\end{align}$
Hence, the above bi-quadratic expression is our answer.

Note: One may note that here we are getting a bi-quadratic expression as our answer because there were four linear expressions in x and we were asked to take their product. In these types of questions you must be careful about the sign between the two terms. Remember the algebraic relations: $\left( -1 \right)\times 1=-1$ and $\left( -1 \right)\times \left( -1 \right)=1$ to simplify the expression. Proceed serially while taking the product otherwise you may get confused in the multiplication of multiple terms.