Questions: Factor completely. (x-5)^2+8(x-5)+12 Simplify your answer as much as possible.

Solution Steps

To factor the given expression completely, we can use a substitution method. Let \( u = x - 5 \). This transforms the expression into a quadratic form in terms of \( u \). We then factor the quadratic expression and substitute back \( x - 5 \) for \( u \).

Given the expression: \[ (x-5)^{2}+8(x-5)+12 \] Let \( u = x - 5 \). Substituting \( u \) into the expression, we get: \[ u^2 + 8u + 12 \]

Next, we factor the quadratic expression \( u^2 + 8u + 12 \): \[ u^2 + 8u + 12 = (u + 6)(u + 2) \]

Now, substitute back \( u = x - 5 \) into the factored form: \[ (u + 6)(u + 2) = (x - 5 + 6)(x - 5 + 2) = (x + 1)(x - 3) \]

Final Answer