Final answer:
To factor the trinomial 5x² + 36x + 36, we realize it's a perfect square and express it as (5x + 6)². The trinomial is a perfect square because the first and last terms are perfect squares and the middle term is twice the product of their square roots.
Step-by-step explanation:
The student is asking how to factor the trinomial 5x² + 36x + 36. Factoring trinomials typically involves finding two binomials that multiply to give the original trinomial. In this case, the trinomial is a perfect square because the first term (5x²) is a square term, the last term (36) is a square term, and the middle term (36x) is twice the product of the square roots of the first and last terms. Therefore, the factored form is (5x + 6)².
To verify, we can expand the binomial:
- (5x + 6) * (5x + 6)
- = 5x * 5x + 5x * 6 + 6 * 5x + 6 * 6
- = 25x² + 30x + 30x + 36
- = 25x² + 60x + 36
Since 25x² + 60x + 36 simplifies to 5(5x² + 12x + 36), which is the original trinomial divided by 5, our factorization (5x + 6)² is correct.