(4x+6)/(16x^2-36)/(4x^2+22x-42)
To simplify the expression
(4x+6)/(16x^2-36)/(4x^2+22x-42),
we need to simplify both the numerator and the denominator separately, and then divide the numerator by the denominator.
First, let's simplify the numerator (4x+6). There are no common factors to factor out, so we can leave it as is.
Next, let's simplify the denominator (16x^2 - 36). We can factor out a common factor of 4 from both terms:
16x^2 - 36 = 4(4x^2 - 9).
Now let's simplify the second denominator (4x^2 + 22x - 42). We can factor it by looking for two numbers that multiply to -42 and add up to 22. The numbers are 14 and -2:
4x^2 + 22x - 42 = (2x + 14)(2x - 3).
Now, the expression becomes:
(4x+6)/(4(4x^2 - 9))/(2x + 14)(2x - 3).
To divide a fraction by a fraction, we invert the second fraction and multiply:
(4x+6) * (2x + 14)(2x - 3) / [4(4x^2 - 9)].
Now, we can multiply the numerator. First, distribute (4x+6) into (2x + 14)(2x - 3):
(4x+6) * (2x + 14)(2x - 3) = (4x+6) * (4x^2 + 8x - 6).
Now distribute (4x+6) into (4x^2 + 8x - 6):
(4x+6) * (4x^2 + 8x - 6) = 16x^3 + 56x^2 + 8x^2 + 24x - 24x - 36.
Combining like terms, the numerator simplifies to:
16x^3 + 64x^2 - 36.
Therefore, the simplified expression is:
(16x^3 + 64x^2 - 36) / [4(4x^2 - 9)].