Question

factor the following polynomial.
$25x^2 + 30x + 9$
$(?x + quad)^2$

Explanation:

Step1: Recall the perfect square trinomial formula

The perfect square trinomial formula is \(a^{2}+2ab + b^{2}=(a + b)^{2}\). For the polynomial \(25x^{2}+30x + 9\), we first identify \(a^{2}\) and \(b^{2}\).
We can see that \(25x^{2}=(5x)^{2}\), so \(a = 5x\). And \(9 = 3^{2}\), so \(b=3\).

Step2: Check the middle term

Now we check the middle term. The middle term of the perfect square trinomial should be \(2ab\). Let's calculate \(2ab\) with \(a = 5x\) and \(b = 3\).
\(2ab=2\times(5x)\times3 = 30x\), which matches the middle term of the given polynomial \(25x^{2}+30x + 9\).
So we can factor the polynomial as \((5x + 3)^{2}\).

Answer:

The first box (coefficient of \(x\)) is \(5\) and the second box is \(3\), so the factored form is \((5x + 3)^{2}\)