To determine how many solutions the system of equations has, we need to find the points of intersection between the two equations:
[tex]\[
y = 2x + 5
\][/tex]
and
[tex]\[
y = x^3 + 4x^2 + x + 2
\][/tex]
To do this, set the two equations equal to each other:
[tex]\[
2x + 5 = x^3 + 4x^2 + x + 2
\][/tex]
Rearrange the equation to bring all terms to one side, forming a polynomial equation:
[tex]\[
x^3 + 4x^2 + x + 2 - (2x + 5) = 0
\][/tex]
Simplify the equation:
[tex]\[
x^3 + 4x^2 + x + 2 - 2x - 5 = 0
\][/tex]
[tex]\[
x^3 + 4x^2 - x - 3 = 0
\][/tex]
Now, we need to determine the number of roots for the polynomial equation:
[tex]\[
x^3 + 4x^2 - x - 3 = 0
\][/tex]
A cubic polynomial (a polynomial of degree 3) can have up to three real roots. By analyzing or solving this polynomial equation, it turns out to have exactly three real roots. Thus, this system of equations has three solutions where the two curves intersect.
The correct answer is:
D. 3 solutions
