Let's start by simplifying the left side of the equation:
\[ 100x^2 - 36 \]
This expression can be factored using the difference of squares:
\[ 100x^2 - 36 = (10x)^2 - 6^2 = (10x - 6)(10x + 6) \]
Now we rewrite the equation:
\[ (10x - 6)(10x + 6) = 4(5x + ?)(5x - 3) \]
Next, we simplify the right side:
\[ 4(5x + ?)(5x - 3) \]
We need to find the missing term denoted by "?". To do that, let's analyze the comparison of both sides as we expand the right side.
The expression \( (5x + ?)(5x - 3) \) can be expanded:
\[ (5x + ?)(5x - 3) = 25x^2 - 15x + ?(5x) - 3? \] \[ = 25x^2 + (? \cdot 5 - 15)x - 3? \]
Now multiply this entire expression by \( 4 \):
\[ 4(25x^2 + (? \cdot 5 - 15)x - 3?) = 100x^2 + 4(? \cdot 5 - 15)x - 12? \]
Now we need to match this with the left side, which we factored as \( (10x - 6)(10x + 6) \). The coefficient of \( x^2 \) matches (both are \( 100x^2 \)), so we will set the corresponding linear terms on both sides equal.
On the left side, the linear terms are 0 since it is fully factored into two quadratic expressions. Therefore, we have:
\[ 4(? \cdot 5 - 15) = 0 \]
This leads to:
\[ ? \cdot 5 - 15 = 0 \] \[ ? \cdot 5 = 15 \] \[ ? = \frac{15}{5} = 3 \]
Now we can confirm our conclusions:
Substituting \( ? \) back into our original equation gives us:
\[ 4(5x + 3)(5x - 3) \]
We can also check the constant terms:
The constant term from the left side after factoring was \( -36 \).
Calculating the constant from the right side:
\[ 4 \cdot 3 \cdot (-3) = -36 \]
Both sides match, thus confirming our value of \( ? \).
The missing term is:
\[ \boxed{3} \]
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