One of the factors of 4x²+ 9y²+ 12xy−5(2x+ 3y)−14 is :

The correct option is 2x + 3y - 7.

To find the factors of the given algebraic expression, we can use the method of algebraic identities and substitution.

The given expression is:

$4x^2+9y^2+12xy-5(2x+3y)-14$

Step 1: Recognize the perfect square trinomial.
Notice the first three terms: 4x² + 9y² + 12xy.
We can rewrite these terms using the algebraic identity ${(a+b)}^2=a^2+2ab+b^2$:

$4x^2={\left(2x\right)}^2$

$9y^2={\left(3y\right)}^2$

$12xy=2·\left(2x\right)·\left(3y\right)$

Therefore, we have:

$4x^2+9y^2+12xy={(2x+3y)}^2$

Step 2: Rewrite the original expression.
Substitute ${(2x+3y)}^2$ back into the original expression:

${(2x+3y)}^2-5(2x+3y)-14$

Step 3: Simplify by substitution.
Let $u=2x+3y$. Now, the expression simplifies to a simple quadratic in terms of $u$:

$u^2-5u-14$

Step 4: Factor the quadratic expression.
To factor $u^2-5u-14$, we look for two numbers whose product is -14 and whose sum is -5.
These two numbers are -7 and +2 because:

$(-7)\times 2=-14$

$-7+2=-5$

Splitting the middle term and factoring:

$u^2-5u-14=(u-7)(u+2)$

Step 5: Substitute back $u=2x+3y$.
Replacing $u$ with $2x+3y$ gives the complete factorization:

$(2x+3y-7)(2x+3y+2)$

Hence, the factors of the given expression are (2x + 3y - 7) and (2x + 3y + 2).
Comparing with the given choices, 2x + 3y - 7 is one of the factors.