Certainly! Let's find the product of the given expressions [tex]\( (9t - 4) \)[/tex] and [tex]\( (-9t - 4) \)[/tex].
First, let's write down the expressions:
[tex]\[ (9t - 4)(-9t - 4) \][/tex]
To find the product, we use the distributive property (also known as the FOIL method for binomials):
[tex]\[ (9t - 4)(-9t - 4) = (9t)(-9t) + (9t)(-4) + (-4)(-9t) + (-4)(-4) \][/tex]
Now, let's calculate each term individually:
1. The product of the first terms:
[tex]\[ (9t)(-9t) = -81t^2 \][/tex]
2. The product of the outer terms:
[tex]\[ (9t)(-4) = -36t \][/tex]
3. The product of the inner terms:
[tex]\[ (-4)(-9t) = 36t \][/tex]
4. The product of the last terms:
[tex]\[ (-4)(-4) = 16 \][/tex]
Next, we combine all these terms:
[tex]\[ -81t^2 + (-36t + 36t) + 16 \][/tex]
Notice that the [tex]\( -36t \)[/tex] and [tex]\( 36t \)[/tex] terms cancel each other out:
[tex]\[ -81t^2 + 0t + 16 \][/tex]
So, the expression simplifies to:
[tex]\[ -81t^2 + 16 \][/tex]
Therefore, the product of [tex]\( (9t - 4)(-9t - 4) \)[/tex] is:
[tex]\[ 16 - 81t^2 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{16 - 81t^2} \][/tex]
