\( \left. \begin{array} { l } { g ^ { 2 } - 5 g - 6 } \\ { = ( g - 3 ) ( g - 2 ) } \end{array} \right. \)

We start with the expression \[ g^2 - 5g - 6 = (g-3)(g-2). \] **Step 1. Expand the Right-Hand Side** Expand \((g-3)(g-2)\) by multiplying the factors: \[ (g-3)(g-2) = g^2 - 2g - 3g + 6 = g^2 - 5g + 6. \] **Step 2. Compare Both Sides** The left-hand side is \[ g^2 - 5g - 6, \] and the right-hand side (after expansion) is \[ g^2 - 5g + 6. \] Since the quadratic and linear terms match, compare the constant terms: - Left-hand side constant: \(-6\) - Right-hand side constant: \(+6\) These constants are not equal. **Step 3. Conclude the Evaluation** Setting the two sides equal gives: \[ g^2 - 5g - 6 = g^2 - 5g + 6. \] Subtract \(g^2 - 5g\) from both sides: \[ -6 = 6, \] which is a contradiction. This shows that the two expressions are not equal. **Step 4. Correct Factorization of \(g^2 - 5g - 6\)** To factor \(g^2 - 5g - 6\), we need two numbers that multiply to \(-6\) (the constant term) and add to \(-5\) (the coefficient of \(g\)). The pair \(-6\) and \(1\) meets these conditions: \[ -6 \times 1 = -6 \quad \text{and} \quad -6 + 1 = -5. \] Thus, the correct factorization is: \[ g^2 - 5g - 6 = (g-6)(g+1). \] **Conclusion** The given equation \[ g^2-5g-6 = (g-3)(g-2) \] does not hold because \[ (g-3)(g-2)=g^2-5g+6, \] which differs from \(g^2-5g-6\). The correct factorization of \(g^2-5g-6\) is \[ (g-6)(g+1). \]