Let f(x) = 9x^2 + 24x + 16

We notice that the function is a complete square :

Then the function has two similar factors which are the square roots of first and last terms:

**==> f(x)= (3x+4)^2**

Let us check the answer:

==> f( x) = (3x)^2 + 2*3x*4 + 4^2

= 9x^2 + 24x + 16

Also we can use the formula to find the roots and then factor:

The roots = (-b + sqrt( b^2 - 4ac) /2 = - 4/3

==> the factor is ( x+ 4/3) ==>multiply by 3:==> (3x + 4 )

**Then the factor is ( 3x+4) * ( 3x+ 4) = ( 3x+ 4)^2**

We can use the fact that any polynomial can be written as a product of linear factors:

ax^n + bx^(n-1) + ... = a(x - x1)(x - x2)*....*(x - xn)

x1,x2,x3....,xn are the roots of the polynomial.

In this case, the polynomial is a quadratic and it could have 2 roots.

We'll get the roots applying quadratic formula:

x1 = [-24 + sqrt(24^2 - 4*9*16)]/2*9

x1 = (-24 + sqrt(576 - 576))/18

x1 = -24/18

x1 = -8/6

x1 = -4/3

x2 = -4/3

Since the discriminant is zero, the equation has 2 real equal roots and the factorization will be:

9X^2 + 24X + 16 = 9(x - x1)(x - x2)

9X^2 + 24X + 16 = 9(x + 4/3)^2

We'll factorize by 1/9:

9X^2 + 24X + 16 = (9/9)(3x + 4)^2

**9X^2 + 24X + 16 = (3x + 4)^2**

9x^2+24x+16.

Wecan factor this by grouping.

The middle term 24x = 12x+12. Also 12*12 = prooduct of end terms.

Therefore we rewrite the given equation as:

9x^2+12x+12x+16.

= 3x(3x+4)+4(3x+4).

= (3x+4)(3x+4).

Therefore the given expression 9x^2+24x+16 = (3x+4)^2.