What are the linear and quadratic factors of the expression 3x4+7x3+4x2?(1 point)%0D%0AResponses%0D%0A%0D%0Ax2(3x+4)(x+1)%0D%0Ax squared left parenthesis 3 x plus 4 right parenthesis left parenthesis x plus 1 right parenthesis%0D%0A%0D%0A(3x+4)(x+1)%0D%0Aleft parenthesis 3 x plus 4 right parenthesis left parenthesis x plus 1 right parenthesis%0D%0A%0D%0Ax2(3x+4)(x+1)(x+1)%0D%0Ax squared left parenthesis 3 x plus 4 right parenthesis left parenthesis x plus 1 right parenthesis left parenthesis x plus 1 right parenthesis%0D%0A%0D%0Ax2(3x−4)(x−1)

Answer 1

To factor the expression \( 3x^4 + 7x^3 + 4x^2 \), we can start by factoring out the greatest common factor (GCF) from the polynomial.

1. The GCF of the terms \( 3x^4 \), \( 7x^3 \), and \( 4x^2 \) is \( x^2 \).
2. Factoring out \( x^2 \) gives: \[ x^2(3x^2 + 7x + 4) \]

Next, we need to factor the quadratic expression \( 3x^2 + 7x + 4 \). We can use the method of factoring by grouping, or look for two numbers that multiply to \( 3 \times 4 = 12 \) and sum to \( 7 \). The numbers \( 3 \) and \( 4 \) work, so we rewrite the middle term: \[ 3x^2 + 3x + 4x + 4 \] Next, we group the terms: \[ (3x^2 + 3x) + (4x + 4) \] Factoring each group, we get: \[ 3x(x + 1) + 4(x + 1) \] Now we factor out the common term \( (x + 1) \): \[ (3x + 4)(x + 1) \]

Putting this all together, we have: \[ x^2(3x + 4)(x + 1) \]

Thus, the linear and quadratic factors of the expression \( 3x^4 + 7x^3 + 4x^2 \) are: \[ \boxed{x^2(3x + 4)(x + 1)} \]