factoring completely:

36x^2 + 60x +25

36x^2 + 60x +25

=(6x)^2+2*5*(6x)+5^2

Does that look familiar?

To factor the quadratic expression completely, you can look for two binomial factors that multiply together to give you the original expression.

Given the expression: 36x^2 + 60x + 25

Step 1: Check if the quadratic expression can be factored using the perfect square trinomial form: a^2 + 2ab + b^2 = (a + b)^2.

In this case, a = 6x and b = 5.

Step 2: Square the value of a, which is 6x, and square the value of b, which is 5.

(6x)^2 = 36x^2
(5)^2 = 25

Step 3: Compare the resulting expression from step 2, which is 36x^2 + 25, with the original expression: 36x^2 + 60x + 25

If the resulting expression matches the original expression, then the quadratic expression can be factored.

Step 4: Replace the middle term of the original expression, 60x, with the two corresponding terms from step 3, which are 2ab: (2 * 6x * 5).

36x^2 + 60x + 25 = 36x^2 + 2(6x)(5) + 25

Step 5: Factor by grouping. Group the first two terms and the last two terms separately.

36x^2 + 2(6x)(5) + 25 = (36x^2 + 2(6x)(5)) + 25

Step 6: Factor out the greatest common factor from each group.

36x^2 + 2(6x)(5) = 2(6x)(6x + 5) = 12x(6x + 5)

Step 7: Combine the factored groups.

12x(6x + 5) + 25 = 12x(6x + 5) + 25

So, the completely factored form of the quadratic expression 36x^2 + 60x + 25 is 12x(6x + 5) + 25.