Question

Factor the expression. Tell which special product factoring pattern you used. $$-27 t^{2}-18 t-3$$

Short answer

The factored form of the expression \(-27 t^{2}-18 t-3\) is \(-3(3t+1)^2\), using the pattern of perfect squares for factoring.

Step-by-step solution

Identify Common Factors

The first step is to identify if there are common factors between each term. The common factor for \(-27t^{2}\), \(-18t\), and \(-3\) is \(-3\). Dividing each term by -3 simplifies the trinomial to \(9t^{2} + 6t + 1\).

Check for Factoring Patterns

Next, check if \(9t^{2} + 6t + 1\) fits any special factoring patterns. Upon inspection, we can see that it fits the pattern \((a+b)^2 = a^2 + 2ab + b^2\). We deduce that a could be \(3t\) and b could be 1 because \( (3t)^2 = 9t^{2}\) and \(2 * (3t) * 1 = 6t\), this fits all terms in our trinomial.

Factoring the Expression

Given that our values of a and b fit into the pattern \((a + b)^2\), we can say that the factored form of \(9t^{2} + 6t + 1\) is \((3t + 1)^2\). Since we factored out -3 initially, the actual answer is \(-3(3t+1)^2\).