Find all integral values of a for which the quadratic expression `(x - a) (x - 10) + 1` can be factored as a product `(x + alpha) (x + beta)` of two factors and `alpha,beta in I.`

Explanation

To solve the problem, we need to find all integral values of \( a \) for which the quadratic expression \( (x - a)(x - 10) + 1 \) can be factored as a product \( (x + \alpha)(x + \beta) \) where \( \alpha \) and \( \beta \) are integers.

### Step 1: Expand the quadratic expression
We start by expanding the given expression:

\[ (x - a)(x - 10) + 1 = x^2 - (a + 10)x + (10a) + 1 \]
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