If p and q are integers and lie between –5 and –1, both inclusive, then the number of equations of the form x2 – px – q = 0 having real and equal roots is:

The equation x^2 - px - q = 0 will have real and equal roots if the discriminant (b^2 - 4ac) is equal to zero. In this case, p and q are integers and lie between -5 and -1, both inclusive.

Step 1: Identify a, b, and c from the equation. In this case, a = 1, b = -p, and c = -q.

Step 2: Set the discriminant equal to zero. This gives us (-p)^2 - 41(-q) = 0.

Step 3: Simplify the equation. This gives us p^2 + 4q = 0.

Step 4: Since p and q are integers between -5 and -1, we can substitute these values into the equation to find the number of solutions.

The possible values for p are -5, -4, -3, -2, -1 and for q are -5, -4, -3, -2, -1.

Step 5: Substitute each possible value of p and q into the equation p^2 + 4q = 0 and count the number of times the equation is satisfied.

After doing this, you will find that there are 5 solutions to the equation, corresponding to the pairs (p, q): (-1, -1), (-2, -1), (-3, -4), (-4, -4), (-5, -6). However, since q cannot be -6, the pair (-5, -6) is not valid.

So, there are 4 equations of the form x^2 - px - q = 0 having real and equal roots.