Example

Factoring 30x2−53xy−21y230x^2 - 53xy - 21y^2

Factor the trinomial 30x2−53xy−21y230x^2 - 53xy - 21y^2 by systematically testing combinations of factors. List the factor pairs for the leading term 30x230x^2 (such as 3x⋅10x3x \cdot 10x) and the last term −21y2-21y^2. Because the last term is negative, its factor pairs must have opposite signs (such as yy and −21y-21y). Test these combinations to find the pair whose inner and outer products sum to the middle term, −53xy-53xy. By selecting the binomials (3x+y)(3x + y) and (10x−21y)(10x - 21y), the outer product is 3x(−21y)=−63xy3x(-21y) = -63xy and the inner product is y(10x)=10xyy(10x) = 10xy. Their sum is −63xy+10xy=−53xy-63xy + 10xy = -53xy, which is exactly the middle term. Therefore, the completely factored form of the expression is (3x+y)(10x−21y)(3x + y)(10x - 21y).

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Updated 2026-06-17

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