Answer
the solution set is\[\left\{ -8,-1 \right\}\].
Work Step by Step
Step 1: Shift all nonzero terms to left side and obtain zero on the other side.
For this, add 8 both sides as follows:
\[{{x}^{2}}+9x+8=-8+8\]
This implies that,
\[{{x}^{2}}+9x+8=0\]
So, after shifting all nonzero terms to the left side, the above equation becomes: \[{{x}^{2}}+9x+8=0\]
Step 2:Find the factor of the above equation.
Consider the equation\[{{x}^{2}}+9x+8=0\].
Factorize it as follows:
\[\begin{align}
& {{x}^{2}}+9x+8=0 \\
& {{x}^{2}}+8x+x+8=0 \\
& x\left( x+8 \right)+1\left( x+8 \right)=0 \\
& \left( x+1 \right)\left( x+8 \right)=0
\end{align}\]
Steps 3 and 4: Set each factor equal to zero and solve the resulting equation:
From step 2, \[\left( x+1 \right)\left( x+8 \right)=0\].
By the zero product principal, either \[\left( x+1 \right)=0\]or\[\left( x+8 \right)=0\].
Now \[\left( x+1 \right)=0\] implies that\[x=-1\] and \[\left( x+8 \right)=0\]implies that\[x=-8\].
Step 5: Check the solution in the original equation.
Check for\[x=-1\]. So consider,
\[\begin{align}
& {{x}^{2}}+9x+8=0 \\
& {{\left( -1 \right)}^{2}}+9\left( -1 \right)+8=0 \\
& 1-9+8=0 \\
& 0=0
\end{align}\]
And, check for\[x=-8\].
\[\begin{align}
& {{x}^{2}}+9x+8=0 \\
& {{\left( -8 \right)}^{2}}+9\left( -8 \right)+8=0 \\
& 64-72+8=0 \\
& 0=0
\end{align}\]
Hence, the solution set is\[\left\{ -8,-1 \right\}\].
