How do you solve \[\left( 4k+5 \right)\left( k+1 \right)=0\]?

Hint: This type of problem is based on the concept of quadratic equations. First, we have to consider the given equation with variable k. Since the equation is represented as the product of two terms which is equal to 0, we get that 4k+5=0 and k+1=0. Now, subtract 5 from the first expression and then divide by 4 to obtain the final solution. Now solve k+1=0 by subtracting 1 from both the sides. Thus, we get the values of k which is the required answer.

Complete step-by-step solution:
According to the question, we are asked to solve \[\left( 4k+5 \right)\left( k+1 \right)=0\].
We have been given the quadratic equation is \[\left( 4k+5 \right)\left( k+1 \right)=0\]. ---------(1)
Since equation (1) is a product of two functions with variable k which is equal to 0, we get that the two functions are equal to 0.
Therefore, \[4k+5=0\] and \[k+1=0\].
First, let us consider 4k+5=0. ---------(2)
Subtract 5 from both the sides of the equation (2). We get
4k+5-5=0-5
We know that the terms with the same magnitude and opposite sign cancel out.
Therefore, 4k=-5.
Now, divide the whole obtained expression by 4. We get
\[\dfrac{4k}{4}=\dfrac{-5}{4}\]
Cancelling out the common term, that is 4, we get
\[k=\dfrac{-5}{4}\].
Now, let us consider k+1=0. -----------(3)
Let us subtract 1 from both the sides of the equation (3). We get,
k+1-1=0-1
We know that the terms with the same magnitude and opposite sign cancel out.
Therefore, k=-1.
Hence, the values of k in the equation \[\left( 4k+5 \right)\left( k+1 \right)=0\] are \[\dfrac{-5}{4}\] and -1.

Note:We can check whether the obtained values of k are correct or not by substituting these values in the given equation \[\left( 4k+5 \right)\left( k+1 \right)=0\].
Consider \[k=\dfrac{-5}{4}\]. Now, substitute the value of k in \[\left( 4k+5 \right)\left( k+1 \right)\]. We get,
\[\left( 4k+5 \right)\left( k+1 \right)=\left( 4\left( \dfrac{-5}{4} \right)+5 \right)\left( \left( \dfrac{-5}{4} \right)+1 \right)\]
\[\Rightarrow \left( 4k+5 \right)\left( k+1 \right)=\left( -5+5 \right)\left( \left( \dfrac{-5}{4} \right)+1 \right)\]
\[\Rightarrow \left( 4k+5 \right)\left( k+1 \right)=\left( 0 \right)\left( \left( \dfrac{-5}{4} \right)+1 \right)\]
\[\therefore \left( 4k+5 \right)\left( k+1 \right)=0\]
Now, consider k=-1. Substitute the value of k in \[\left( 4k+5 \right)\left( k+1 \right)\]. We get
\[\left( 4k+5 \right)\left( k+1 \right)=\left( 4\left( -1 \right)+5 \right)\left( -1+1 \right)\]
\[\Rightarrow \left( 4k+5 \right)\left( k+1 \right)=\left( 14+5 \right)\left( 0 \right)\]
\[\therefore \left( 4k+5 \right)\left( k+1 \right)=0\]
Hence, the obtained values of k are verified.