If the quadratic relationship x^2 + x = 19 holds for a real number x, what is the exact value of the expression (x + 5)^2 + 1 / (x + 5)^2?

Introduction / Context:
This question involves transforming a given quadratic condition into a new expression involving a shifted variable. You must use algebraic identities carefully to compute a sum of a square and its reciprocal square without explicitly solving for x. This technique is common in aptitude exams where exact numeric solutions for x are not required.

Given Data / Assumptions:

• x satisfies the equation x^2 + x = 19.
• x is a real number.
• We must find (x + 5)^2 + 1/(x + 5)^2.
• The expression is defined, so x + 5 is non zero.

Concept / Approach:
Introduce a new variable t = x + 5 to simplify the expression. Rewrite the original equation in terms of t, which leads to a quadratic equation in t. From that quadratic, deduce a simple relation between t and 1/t. Then use the identity (t + 1/t)^2 = t^2 + 2 + 1/t^2 to obtain t^2 + 1/t^2, which is exactly the required expression.

Step-by-Step Solution:
Let t = x + 5, so x = t − 5.Substitute x = t − 5 into x^2 + x = 19 to express in terms of t.Compute (t − 5)^2 + (t − 5) = 19, which expands to t^2 − 10t + 25 + t − 5 = 19.Simplify: t^2 − 9t + 20 = 19, so t^2 − 9t + 1 = 0.Divide by t (t is non zero): t − 9 + 1/t = 0, giving t + 1/t = 9.Square both sides: (t + 1/t)^2 = t^2 + 2 + 1/t^2 = 81.Therefore, t^2 + 1/t^2 = 81 − 2 = 79. Since t = x + 5, (x + 5)^2 + 1/(x + 5)^2 = 79.

Verification / Alternative check:
We can solve x^2 + x − 19 = 0 directly using the quadratic formula, get two real roots, then compute (x + 5)^2 + 1/(x + 5)^2 numerically for each root. For both roots, the value of the expression is 79. This numerical check confirms that the identity based method is correct and that the expression is independent of which root of the quadratic is chosen.

Why Other Options Are Wrong:

• 77 and 81 are close but arise from small arithmetic mistakes, such as forgetting to subtract 2 after squaring.
• 83 and 75 are further away and suggest incorrect manipulation of the quadratic in t or an error in the t + 1/t step.
• Only 79 agrees exactly with the identity t^2 + 1/t^2 = 81 − 2.

Common Pitfalls:

• Forgetting to define t = x + 5 and instead trying to handle the expression in terms of x directly, which is more cumbersome.
• Incorrectly expanding (t − 5)^2 or miscombining the linear terms when forming the quadratic in t.
• Misusing the identity for (t + 1/t)^2 and forgetting the extra +2 term before isolating t^2 + 1/t^2.

Final Answer:
79