What are the solutions for the system of equations x + y = 5 and y + 1 = 3x² + 2x?

To find the solutions for the system of equations given by ( x + y = 5 ) and ( y + 1 = 3x^2 + 2x ), we need to solve these equations simultaneously.

Starting with the first equation ( x + y = 5 ), we can express ( y ) in terms of ( x ):

[

y = 5 - x.

]

Now, we substitute this expression for ( y ) into the second equation ( y + 1 = 3x^2 + 2x ):

[

(5 - x) + 1 = 3x^2 + 2x.

]

Simplifying this gives:

[

6 - x = 3x^2 + 2x.

]

Rearranging terms to bring everything to one side results in:

[

3x^2 + 3x - 6 = 0.

]

Dividing the entire equation by 3 simplifies it to:

[

x^2 + x - 2 = 0.

]

Next, we factor this quadratic equation:

[

(x + 2)(x - 1) = 0.

]

Setting