To factor the trinomial 4x^2y+44xy-12y completely, we can first factor out the greatest common factor of the three terms, which is 4y. This gives us:
4y( x^2 + 11x - 3)
Now we need to factor the quadratic expression x^2 + 11x - 3. We can use the quadratic formula or factoring by grouping to factor this expression. Factoring by grouping involves finding two numbers that multiply to give the constant term (-3) and add to give the coefficient of the middle term (11). These numbers are 11 and -3, so we can rewrite the expression as:
x^2 + 11x - 3 = x^2 + 11x + (-3x) - 3 = (x^2 - 3x) + (11x - 3) = x(x - 3) + 3(11x - 1)
Therefore, the fully factored form of the trinomial 4x^2y+44xy-12y is:
4y(x - 3)(x + 11)