Find the parameter a that belongs to R if parabola y = (a + 1)*x^2 + a*x + 3 and the line y = x + 1 have two distinct points in common.

At the point of intersection of y = (a + 1)*x^2 + a*x + 3
and y = x + 1, the y- values are equal. Therefore equating them we
get:

(a + 1)*x^2 + a*x + 3 = x +
1

=> (a + 1)*x^2 + a*x - x + 3 - 1 =
0

=> (a + 1)*x^2 + (a -1 ) *x + 2 =
0

Now according to the initial condition there are two
distinct points in common, therefore (a-1)^2 - 4*(a+1)*2 >
0

=> a^2 - 2a +1 - 8a - 8 >
0

For a^2 - 10a - 7 =0

the
roots are [10 + sqrt( 100 + 28)] /2 and [10 - sqrt ( 100 +
28)]/2

or 5 + 4 sqrt 2 and 5 - sqrt
2.

Therefore as a^2 - 2a +1 - 8a - 8 should be greater than
0, a should lie either below 5 - 4 sqrt 2 or above 5 + 4 sqrt
2.

a can be either below 5 - 4 sqrt 2 or
above 5 + 4 sqrt 2.