1. **Stating the problem:** We want to understand how to factor trinomials and identify conditions for perfect square trinomials. 2. **Factoring a trinomial:** A trinomial of the form $a^2 + bx + c$ can be factored into $(a + m)(a + n)$ where $m$ and $n$ satisfy: - $m + n = b$ - $mn = c$ 3. **Example for the trinomial $a^2 - 12a + c$:** - Here, $b = -12$. - We look for two numbers $m$ and $n$ such that $m + n = -12$ and $mn = c$. - For instance, if $m = -6$ and $n = -6$, then $m + n = -12$ and $mn = 36$, so $c = 36$. - This matches Response 1: $a=1$ and $c=36$. 4. **Perfect square trinomial condition:** A perfect square trinomial is of the form $(pa + q)^2 = p^2a^2 + 2pqa + q^2$. - For $4a^2 + 44a + c$ to be a perfect square, it must match $ (2a + k)^2 = 4a^2 + 4ak + k^2$. - Comparing coefficients: $44a = 4ak ightarrow k = 11$. - Then $c = k^2 = 11^2 = 121$. 5. **Summary:** - To factor trinomials, find two numbers that add to the middle coefficient and multiply to the constant term. - For perfect square trinomials, the middle term must be twice the product of the square roots of the first and last terms. **Final answers:** - For $a^2 - 12a + c$ factored as $(a - 6)^2$, $c = 36$. - For $4a^2 + 44a + c$ to be a perfect square, $c = 121$.