To determine how many solutions the system of equations has, we need to find the points of intersection between the two equations.
The given system of equations is:
[tex]\[
\begin{array}{l}
y = 2x + 5 \\
y = x^3 + 4x^2 + x + 2
\end{array}
\][/tex]
Since both expressions are set equal to [tex]\( y \)[/tex], we can set the right sides of the equations equal to each other:
[tex]\[
2x + 5 = x^3 + 4x^2 + x + 2
\][/tex]
To find the solutions, we need to solve this equation:
[tex]\[
2x + 5 = x^3 + 4x^2 + x + 2
\][/tex]
First, bring all terms to one side of the equation:
[tex]\[
0 = x^3 + 4x^2 + x + 2 - 2x - 5
\][/tex]
Simplify the equation:
[tex]\[
0 = x^3 + 4x^2 - x - 3
\][/tex]
Now, we need to find the roots of the polynomial equation [tex]\( x^3 + 4x^2 - x - 3 = 0 \)[/tex]. The number of roots of a cubic equation corresponds to the number of solutions for [tex]\( x \)[/tex] where the two equations intersect.
This specific polynomial has exactly 3 roots. Therefore, the equation has 3 solutions.
So, the number of solutions to the system of equations is:
[tex]\[
\boxed{3}
\][/tex]
