ANSWER
[tex]10in,13in[/tex]
EXPLANATION
The tabletop is rectangular in shape. The area of a rectangle is the product of its length and width:
[tex]A=L\cdot W[/tex]
The area of the rectangular tabletop is given as:
[tex]A=(x^2+5x+4)in^2[/tex]
Let us write this area as a product of two terms, just like the general formula for the area of a rectangle.
To do this, factorize the quadratic expression above:
[tex]\begin{gathered} A=x^2+4x+x+4 \\ \Rightarrow A=x(x+4)+1(x+4) \\ \Rightarrow A=(x+1)(x+4) \end{gathered}[/tex]
By comparing this to the general formula for the area of a rectangle, we can say that the length and width of the table are:
[tex]\begin{gathered} L=(x+1)in \\ W=(x+4)in \end{gathered}[/tex]
The distance around the table is 46 inches. This represents the perimeter of the tabletop.
The perimeter of a rectangle is:
[tex]P=2(L+W)[/tex]
Hence, we can substitute the length, width, and perimeter of the tabletop into the equation:
[tex]46=2\lbrack(x+1)+(x+4)\rbrack[/tex]
Solve for x in the equation above:
[tex]\begin{gathered} 46=2(2x+5)=4x+10 \\ \Rightarrow4x=46-10=36 \\ x=\frac{36}{4} \\ x=9 \end{gathered}[/tex]
Therefore, substituting the value of x into the equation for the length and width, the actual side lengths of the table are:
[tex]\begin{gathered} \Rightarrow L=9+1=10in \\ \Rightarrow W=9+4=13in \end{gathered}[/tex]
That is the answer.
