By foctorising `x^(2) - 22x + 117 ` , evaluate
` " " ( x^(2) -22x +117)+ (x-13) `

Answer

To solve the problem step by step, we will first factor the quadratic expression \( x^2 - 22x + 117 \) and then evaluate the expression \( (x^2 - 22x + 117) + (x - 13) \).

### Step 1: Factor the quadratic expression

We start with the expression:
\[ x^2 - 22x + 117 \]

We need to find two numbers that multiply to \( 117 \) (the constant term) and add up to \( -22 \) (the coefficient of \( x \)).

After checking the factors of \( 117 \), we find:
- The factors are \( 1 \times 117 \), \( 3 \times 39 \), \( 9 \times 13 \).

Among these, the pair \( 9 \) and \( 13 \) can be used, since:
\[ -9 + (-13) = -22 \quad \text{and} \quad -9 \times -13 = 117 \]

Thus, we can rewrite the quadratic as:
\[ (x - 9)(x - 13) \]

### Step 2: Rewrite the expression

Now we can substitute the factorized form back into the expression we need to evaluate:
\[ (x^2 - 22x + 117) + (x - 13) = (x - 9)(x - 13) + (x - 13) \]

### Step 3: Factor out the common term

Notice that \( (x - 13) \) is a common factor in both terms:
\[ = (x - 13)((x - 9) + 1) \]

### Step 4: Simplify the expression inside the parentheses

Now simplify the expression inside the parentheses:
\[ (x - 9) + 1 = x - 8 \]

So, we have:
\[ = (x - 13)(x - 8) \]

### Final Result

The factorized form of the original expression is:
\[ (x - 13)(x - 8) \]

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