The Berry Method has you copy the coefficient of the first term into the first term of each of the binomial factors. As a second step, you find factor of "a*c" that add to "b", where "a", "b", "c" are the coefficients of ax^2 +bx +c.
So, you start with a factorization that looks like
.. (ax + __)*(ax + __)
with the blanks to be filled by the numbers you determine in the second step.
Finally, you divide this product by "a" in an appropriate way. Often, that will mean you factor "a" from one of the binomials.
291) 5c^2 +13c +6
.. form to fill: (5c + __)*(5c + __)
We're looking for factors of 5*6 = 30 that add to 13.
.. 30 = 1*30 = 2*15 = 3*10 = 5*6 . . . . . . sums of factors are 31, 17, 13, 11.
The numbers we want are 3 and 10. Putting these into the form gives
.. (5c +3)*(5c +10)
.. = 5(5c +3)(c +2)
The factorization we want is
.. 5c^2 +13c +6 = (5c +3)(c +2)
294) 4d^2 +8d -21
.. 4*-21 = -84 = 84*-1 = 42*-2 = 28*-3 = 21*-4 = 14*-6 = 12* -7
.. These factor pairs have sums 83, 40, 25, 17, 8, 5, so our numbers are 14, -6.
.. (4d +14)(4d -6) . . . . form with numbers filled in
.. = 2(2d +7)*2(2d -3) . . . . . remove a factor of 4 by removing a factor of 2 from each binomial
.. = 4(2d +7)(2d -3)
The factorization we want is
.. 4d^2 +8d -21 = (2d +7)(2d -3)
297) 3g^2 -16g -12
.. 3*-12 = -36 = -36*1 = -18*2 = -12*3 = -9*4 = -6*6
.. These factor pairs have sums -35, -16, -9, -5, 0, so the numbers we want are -18 and 2.
The Berry Method form now looks like
.. (3g -18)(3g +2)
.. = 3(g -6)(3g +2)
The factorization we want is
.. 3g^2 -16g -12 = (g -6)(3g +2)
