To factor a perfect square trinomial, follow these steps:

1. Identify a perfect square trinomial:
   The trinomial should be in the form a2x2+2abx+b2a^2x^2+2abx+b^2a2x2+2abx+b2 or a2x2−2abx+b2a^2x^2-2abx+b^2a2x2−2abx+b2, where the first and last terms are perfect squares, and the middle term is twice the product of the square roots of the first and last terms

2. Find the square roots of the first and last terms:
   • The square root of the first term a2x2a^2x^2a2x2 is axaxax.
   • The square root of the last term b2b^2b2 is bbb

3. Check the middle term:
   Confirm that the middle term is ±2abx\pm 2abx±2abx. The sign of the middle term determines whether the binomial factor uses addition or subtraction

4. Write the factored form:
   • If the middle term is positive, factor as (ax+b)2(ax+b)^2(ax+b)2.
   • If the middle term is negative, factor as (ax−b)2(ax-b)^2(ax−b)2

Example

Factor 9x2+30x+259x^2+30x+259x2+30x+25:

• 9x2=(3x)29x^2=(3x)^29x2=(3x)2
• 25=5225=5^225=52
• The middle term 30x=2×3x×530x=2\times 3x\times 530x=2×3x×5
• Since the middle term is positive, the factorization is (3x+5)2(3x+5)^2(3x+5)2

Summary

• Confirm first and last terms are perfect squares.
• Confirm middle term equals twice the product of the square roots of first and last terms.
• Use the sign of the middle term to decide the sign in the binomial.
• Write the factorization as the square of a binomial.

This method reverses the process of squaring a binomial and is a quick way to factor perfect square trinomials