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By using this property of triangle we can write

\[\left( {{x^2} + 2x} \right) + \left( {2x + 3} \right) > {x^2} + 3x + 8 - - - - (i)\]

Now solve for \[x\] we can write:

\[ \left( {{x^2} + 2x} \right) + \left( {2x + 3} \right) > {x^2} + 3x + 8 \\

{x^2} + 4x + 3 > {x^2} + 3x + 8 \\

4x - 3x > 8 - 3 \\

x > 5 - - - - (ii) \\ \]

Now for the other side

\[\left( {{x^2} + 2x} \right) + {x^2} + 3x + 8 > \left( {2x + 3} \right) - - - - (iii)\]

Solving for \[x\]:

\[ \left( {{x^2} + 2x} \right) + {x^2} + 3x + 8 > \left( {2x + 3} \right) \\

2{x^2} + 5x + 8 > 2x + 3 \\

2{x^2} + 5x - 2x > 3 - 8 \\

2{x^2} + 3x + 5 > 0 \\

x = - 0.75 \pm 1.39i \\ \]

The above equation will have the complex values of \[x\] and so can be neglected.

Now check for the third side,

\[\left( {2x + 3} \right) + {x^2} + 3x + 8 > \left( {{x^2} + 2x} \right) - - - - (iv)\]

solving for the value of \[x\]:

\[ \left( {2x + 3} \right) + {x^2} + 3x + 8 > \left( {{x^2} + 2x} \right) \\

{x^2} + 5x + 11 > {x^2} + 2x \\

3x > - 11 \\

x > \dfrac{{ - 11}}{3} \\ \]

The values obtained \[x\] from the three equations (i), (ii), and (iii) we can say that the length of the side of a triangle can never be in negative or in complex form, hence the value of \[x\] the side of the triangle is \[x > 5\].