Replace each of the question marks with a value that will make the polynomial a perfect square trinomial.

1. 4x^2+12x+?
2. ?+90x+81
Could someone show me how to solve these?

Take the first:

factor out the 4

4x^2+12x+?
4( x^2 + 3x + ?)

take half the center term, and square it. half is 3/2, squared is 9/4

4( x^2 + 3x + 9/4)
Put the 4 back in...

4x^2 + 12x+9
(2x+3)(2x+3)

Second:
?+90x+81
Notice the last term is a perfect square, its square root is 9. Twice nine is eighteen.
Now to find the missing term, divide 90 by 18: five.

25x^2 + 90x + 81
(5x+9)(5x+9)

To find the missing value in the given polynomial to make it a perfect square trinomial, you can follow these steps:

1. Identify the coefficient of the middle term in the polynomial.

2. Take half of the coefficient and square it.

3. Replace the missing value with the square obtained in step 2.

4. Factor out any common factors from the polynomial, if necessary.

Let's apply these steps to the first polynomial: 4x^2 + 12x + ?

1. The coefficient of the middle term is 12.

2. Half of 12 is 6, and when squared, it becomes 36.

3. Replace the missing value with 36: 4x^2 + 12x + 36.

4. Since there are no common factors to be factored out, the quadratic is already factored as (2x + 6)(2x + 6), or simplified as (2x + 6)^2.

Now, let's apply the same steps to the second polynomial: ? + 90x + 81.

1. The coefficient of the middle term is 90.

2. Half of 90 is 45, and when squared, it becomes 2025.

3. Replace the missing value with 2025: 2025 + 90x + 81.

4. Again, there are no common factors to be factored out, so the quadratic is already factored as (45x + 9)(45x + 9), or simplified as (45x + 9)^2.